UNIVERSIDAD NACIONAL AUTÓNOMA DE MÉXICO
PROGRAMA DE MAESTRÍA Y DOCTORADO EN CIENCIAS
MATEMÁTICAS Y DE LA ESPECIALIZACIÓN EN ESTADÍSTICA
APLICADA
SAMPLE COPULA PROPERTIES AND WEAK CONVERGENCE OF THE
SAMPLE PROCESS
T E S I S
QUE PARA OPTAR POR EL GRADO DE:
DOCTOR EN CIENCIAS
PRESENTA:
RICARDO HOYOS ARGÜELLES
DIRECTOR DE LA TESIS
DR. JOSÉ MARÍA GONZÁLEZ-BARRIOS MURGUÍA, IIMAS, UNAM
MIEMBROS DEL COMITÉ TUTOR
DRA. MARÍA ASUNCIÓN BEGOÑA FERNÁNDEZ FDEZ, FC, UNAM
DR. ARTURO ERDELY RUIZ, FES ACATLÁN, UNAM
CIUDAD DE MÉXICO A 29 DE NOVIEMBRE DE 2017

Contents
Introduction I
1 Preliminaries 1
1.1 Basic definitions and properties of copulas . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Archimedean copulas and types of dependence . . . . . . . . . . . . . . . . . . . 6
2 Comparison between the empirical copula and the sample d-copula of order m 11
2.1 Definitions and results about the empirical copula . . . . . . . . . . . . . . . . . . 11
2.2 Sample d-copula of order m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3 Simulation study: dimension two . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.4 Simulation study: dimension three . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.5 A method for estimating an adequate m . . . . . . . . . . . . . . . . . . . . . . . 46
3 Moments of some of the random variables associated with the sample copula 61
3.1 Case: dimension two when m divides n . . . . . . . . . . . . . . . . . . . . . . . 67
3.2 Case: dimension three when m divides n . . . . . . . . . . . . . . . . . . . . . . . 77
3.3 Summary of results and generalizations . . . . . . . . . . . . . . . . . . . . . . . 94
3.4 Case: m does not divide n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
4 Convergence results of the sample copula 106
4.1 Weak convergence of empirical process . . . . . . . . . . . . . . . . . . . . . . . 106
4.2 Weak convergence of sample copula process . . . . . . . . . . . . . . . . . . . . . 111
4.3 Simulation study: Gaussian process . . . . . . . . . . . . . . . . . . . . . . . . . 124
Conclusions 128
References 130

Introduction
The sample d-copula of order m is a new way to estimate a d copula, see [24], it is an estimator
which is already a copula, unlike the empirical copula which is only a subcopula.
In this thesis we will study the main two properties of the sample d-copula, that is, a Glivenko-
Cantelli’s theorem and the asymptotic properties of its associated empirical process. The main
objective of this work is to obtain the same results that exist for the empirical copula. The parameter
m, which it is an integer with 2 ≤ m ≤ n, where n is the sample size, is not difficult to estimate
and in most cases it is less than n. Besides the evaluation of the sample copula is fairly simple and
quicker than the empirical copula.
In Chapter 1 we given a review of the basic results of the theory of copulas and we describe the
main definitions needed for the rest of the thesis.
In Chapter 2 we provide definitions and results corresponding to the sample and empirical copula
and we give the proof of a version of the Glivenko-Cantelli’s theorem for the sample d-copula. We
also provide a large number of simulations to compare the approximation of the empirical copula
and the sample d-copula to the real copula based on samples. At the end of this chapter we provide
a method to estimate the order m, with 2 ≤ m ≤ n, of the sample d-copula using the estimated value
of the Sperman’s rho in dimension two, and we also provide a methodology for higher dimensions.
In Chapter 3 we present the behavior of the random variables associated with the counting process
generated by the sample d-copula. We obtain similar results to the ones found in [8] for the two
dimensional case, and we extend the study of the these random variables for higher dimensions.
In Chapter 4 we found the weak convergence of the process generated by the sample d-copula
using the convergence result of the empirical copula process. We obtain the weak convergence to
a Gaussian process, and we evaluate its variance-covariance structure. Finally, in this chapter we
provide several simulations to test the convergence of this process at each point.
In the final remarks we point out the advantages of using the sample d-copula instead of the em-
pirical copula and we also give several possibles extensions and applications of the sample d-copula
for real data.
I
1 Preliminaries
This chapter describes the basic results about copulas used in this work, we provide some exam-
ples related to the Archimedean copulas and the relations between copulas and other dependence
measures. Most of the results presented here can be found in Nelsen’s book [37].
1.1 Basic definitions and properties of copulas
Definition 1.1 Let S 1, S 2 ⊂ R be not empty sets, with R = [−∞,∞] the extended real line. A
function H : S 1 × S 2 −→ R is grounded if there exists a least element a1 ∈ S 1 and a least element
a2 ∈ S 2 such that H(x, a2) = H(a1, y) = 0 for all (x, y) ∈ S 1 × S 2.
Definition 1.2 Let S 1, S 2 ⊂ R be not empty sets, with R = [−∞,∞] the extended real line. Let
B = [x1, x2]×[y1, y2] be a box with vertices in the domain of the function H, we define the H-volume
of B by
VH(B) = H(x2, y2) − H(x2, y1) − H(x1, y2) + H(x1, y1).
Definition 1.3 Let S 1, S 2 ⊂ R be not empty sets and let H : S 1 × S 2 −→ R be a bivariate function.
We say that H is 2-increasing if VH(B) ≥ 0 for every box B with vertices in the domain of the H
function.
Definition 1.4 A two-dimensional subcopula (or 2-subcopula, or briefly subcopula) is a function
C′ with the following properties
1. Dom(C′) = S 1×S 2, where S 1 y S 2 are subsets of I = [0, 1] such that 0, 1 ∈ S 1 and 0, 1 ∈ S 2.
2. C′ is a grounded function and is 2-increasing.
3. For all u ∈ S 1 and for all v ∈ S 2,
C′(u, 1) = u y C′(1, v) = v.
The range of the function C′ is a subset of I = [0, 1].
Definition 1.5 A two-dimensional copula (or 2-copula, or briefly copula) is a 2-subcopula C with
domain equal to I2 = [0, 1]2. That is, a copula C is a function with domain I2 = [0, 1]2 and range
I = [0, 1] that satisfies the following properties
1. For all u, v ∈ I
C(u, 0) = 0 = C(0, v)
and
C(u, 1) = u and C(1, v) = v.
1
2. For all u1, u2, v1, v2 ∈ I such that u1 ≤ u2 and v1 ≤ v2
C(u2, v2) −C(u2, v1) −C(u1, v2) +C(u1, v1) ≥ 0.
Example 1.6 An example of a copula is the function Π2 : [0, 1]2 → [0, 1] defined by Π2(u, v) =
u · v, this function satisfies the above conditions. If we define W2(u, v) = max{0, u + v − 1} and
M2(u, v) = min{u, v} then M2 and W2 are also copulas which satisfy that for every subcopula or
copula C we have that
W2(u, v) ≤ C(u, v) ≤ M2(u, v) for every u, v ∈ [0, 1].
The copulas W2 and M2 are called the lower and upper Fréchet-Hoeffding bounds
Lemma 1.7 For every C subcopula and for every u1, u2, v1, v2 ∈ [0, 1] we have that
|C(u1, v1) −C(u2, v2)| ≤ |u1 − u2| + |v1 − v2|,
which guarantees that C is uniformly continuous (Lipschitz).
Following, the above definitions extends to the case n-dimensional.
Definition 1.8 Let S 1, · · · , S n ⊂ R be not empty sets, with R = [−∞,∞] the extended real line.
A function H : S 1 × · · · × S n −→ R is grounded if there exists a least element ak ∈ S k for every
k ∈ {1, · · · , n}, such that H(t) = 0 for all t ∈ Dom(H), with tk = ak, for at least one k ∈ {1, · · · , n}.
Definition 1.9 Let S 1, · · · , S n ⊂ R be not empty sets, with R = [−∞,∞] the extended real line. Let
H : S 1 × · · · × S n −→ R be a function and let B = [a, b] be a n-box with vertices in the domain of
the H, we define the H-volume of B by
VH(B) =
∑
sgn(c)H(c), (1)
where the sum is considered on all vertices c of B and we define
sgn(c) =
{
1 if ck = ak for an even number of values k
−1 if ck = ak for an odd number of values k.
Example 1.10 Let H : R
3 −→ R be a function and let B = [x1, x2] × [y1, y2] × [z1, z2] be a 3-box.
Then the H-volume of B is given by
VH(B) = H(x2, y2, z2) − H(x2, y2, z1) − H(x2, y1, z2) − H(x1, y2, z2)
+H(x2, y1, z1) + H(x1, y2, z1) + H(x1, y1, z2) − H(x1, y1, z1).
2
Definition 1.11 Let S 1, · · · , S n ⊂ R be not empty sets and let H : S 1×· · ·×S n −→ R be a n-variate
function. We say that H is n-increasing if VH(B) ≥ 0 for all n-box B with vertices in the domain of
the function H.
Definition 1.12 Let S 1, · · · , S n ⊂ R be not empty sets and let H : S 1 × · · · × S n −→ R be a
n-variate function. If every set S k has a maximum element bk, k ∈ {1, · · · , n}, we say that the H
function has marginals and we define the k-th marginal function of one dimension of H, denoted
by Hk : S k −→ R by
Hk(x) = H(b1, · · · , bk−1, x, bk+1, · · · , bn).
for all x ∈ S k. We can define marginals of higher order that one (k-marginals), setting fewer
positions in the domain of H.
Definition 1.13 A n-dimensional subcopula (or n-subcopula) is a function C′ with the following
properties
1. Dom(C′) = S 1 × · · · × S n, with S k ⊂ I such that 0, 1 ∈ S k for every k ∈ {1 · · · , n}.
2. C′ is grounded and is n-increasing.
3. C′ has marginals and each marginal C′
k
, for every k ∈ {1 · · · , n}, satisfies
C′k(u) = u para toda u ∈ S k.
The range of the C′ function is a subset of I = [0, 1].
Definition 1.14 A n-dimensional copula (or n-copula) is a n-subcopula C with domain equal to
In = [0, 1]n. That is, a n-copula C is a function with domain In and range I that satisfies the
following properties
1. For all u ∈ In
C(u) = 0 if at least one coordinate of u is 0
and, if every coordinate of u are 1 except uk, then C(u) = uk.
2. For all a, b ∈ In, such that a ≤ b, that is, ak ≤ bk for every k ∈ {1, · · · , n}, we have that
VC([a, b]) ≥ 0.
Remark 1.15 For all 2 ≤ k < n, every k−marginal function of C is a k-copula.
3
Lemma 1.16 Let C : [0, 1]d → [0, 1] be a d-copula. We define for every u = (u1, . . . , ud) ∈ Id =
[0, 1]d
Wd(u) = max(0,
d
∑
i=1
ui − d + 1) and Md(u) = min(u1, . . . , ud).
Then we have that for every d-copula or d-subcopula C ,
Wd(u) ≤ C(u) ≤ Md(u). (2)
In this case Md is always a d-copula. However, Wd is never a d-copula for d > 2, because the
volume of the box R = [1/2, 1]d is given by VWd (R) = 1 − (d/2) < 0 for every d > 2. However, the
inequality (2) is sharp because for every u ∈ [0, 1]d there exists a d-copula C depending on u, such
that the left inequality in (2) becomes an equality.
Lemma 1.17 For every u, v ∈ [0, 1]d and every d-subcopula C we have that
|C(u) −C(v)| ≤
d
∑
i=1
|ui − vi|.
Again any d-copula C is a continuous (Lipschitz) function.
Theorem 1.18 (Sklar’s Theorem) Let H be a joint d-distribution function for d ≥ 2 with margins
F1, F2, . . . , Fd. Then there exists a d-copula C such that for every (x1, x2, . . . , xd) ∈ RI d,
H(x1, x2, . . . , xd) = C(F1(x1), F2(x2), . . . , Fd(xd)). (3)
If F1, F2, . . . , Fd are continuous, then C is unique; otherwise, C is uniquely determined on Ran(F1)×
Ran(F2) × · · · × Ran(Fd). Conversely, if C is a d-copula and F1, F2, . . . , Fd are distribution func-
tions, then the function H defined in equation (3) is a joint d-distribution function.
Definition 1.19 Given a copula C and a, b ∈ [0, 1] we define the horizontal section of C at a by
ha(t) = C(t, a) and the vertical section of C at b by vb(t) = C(b, t) for every t ∈ [0, 1].
Remark 1.20 Using equation (1) we can see that the functions ha(t) and vb(t) are increasing func-
tions.
Definition 1.21 We define the diagonal section of C by
δC(t) = C(t, t) for every t ∈ [0, 1].
4
Remark 1.22 We can see that δC(t) is also an increasing function and, in the case of d-copulas
C, with d > 2, we have equivalent definitions for each coordinate by fixing d − 1 coordinates, and
defining δC(t) = C(t, t, . . . , t).
Remark 1.23 Given a d-copula C for any (v1, . . . vi−1, vi+1, . . . , vd) ∈ [0, 1]d−1 we have that
∂C
∂ui
(v1, . . . , vi−1, ui, vi+1, . . . , vd)
exists for almost every ui ∈ [0, 1] and for every i ∈ {1, . . . , d}. In fact,
0 ≤ ∂
∂ui
C(v1, . . . , vi−1, ui, vi+1, . . . , vd) ≤ 1.
The last inequality follows from the Lipschitz continuity of C. Furthermore the partials are defined
and nondecreasing almost everywhere (a.e.) on [0, 1]d−1 with respect to Lebesgue measure.
The mixed partials of C also exist almost everywhere with respect to Lebesgue measure. In fact,
c(u1, . . . , cd) =
∂d
∂u1 · · · ∂ud
C(u1, . . . , ud)
also exists a.e. and it is defined as the density function c of the distribution function C.
For example, in the case of Πd we have that its density is given simply by πd(u1, . . . , ud) = 1 for
every (u1, . . . , ud) ∈ [0, 1]d.
Definition 1.24 We define the support of a d-copula C by
supp(C) = {u ∈ [0, 1]d |VC(Rr) > 0 for every r > 0},
where Rr = [u − r, u + r] ⊂ [0, 1]d and r = (r, r, . . . , r) (d times).
Example 1.25 We have that the support of the copula M2 is given by the main diagonal D =
{(u, v) ∈ [0, 1]2 | u = v}. Similarly the support of the copula W2 is the secondary diagonal D1 =
{(u, v) ∈ [0, 1]2 | u + v = 1}.
Definition 1.26 For any d-copula C we define
Ca.c.(u1, . . . , ud) =
∫ u1
0
· ·
∫ ud
0
∂d
∂v1 · · · ∂vd
C(v1, . . . , vd)dvd . . . dv1
5
where a.c. stands for absolutely continuous. Also define
Cs.(u1, . . . , ud) = C(u1, . . . , ud) −Ca.c.(u1, . . . , ud),
where s. stands for singular.
If C = Ca.c. we say that C is absolutely continuous, if C = Cs. we say C is singular, in any other
case C is hybrid.
We can observe that Cs.(1, . . . , 1) is the measure of the singular part if it exists.
1.2 Archimedean copulas and types of dependence
Definition 1.27 Let ϕ : [0, 1] → [0,∞] be a continuous, strictly decreasing, convex function, that
is, if u, v ∈ [0, 1] and 0 ≤ α ≤ 1 then ϕ(αu + (1 − α)v) ≤ αϕ(u) + (1 − α)ϕ(v), such that ϕ(1) = 0,
in this case we will call ϕ a generator. Then, if we define
ϕ[−1](t) =
{
ϕ−1(t) if 0 ≤ t ≤ ϕ(0)
0 if ϕ(0) ≤ t ≤ ∞,
called pseudo-inverse of ϕ, and we define
C(u, v) = ϕ[−1](ϕ(u) + ϕ(v)) for every (u, v) ∈ [0, 1]2. (4)
Then C is always a copula with generator ϕ, ϕ is strict if ϕ(0) = ∞ and non-strict if ϕ(0) < ∞.
The copulas given in equation (4) are called Archimedean copulas.
Lemma 1.28 Let ϕ be a generator of C as in equation (4). Then
i) C is symmetric, that is, C(u, v) = C(v, u) for every (u, v) ∈ [0, 1]2.
ii) C is associative, that is, C(C(u, v),w) = C(u,C(v,w)) for every u, v,w ∈ [0, 1].
iii) If c > 0 then ψ = c · ϕ is also a generator of C.
Theorem 1.29 C is an Archimedean copula if and only if C is an associative copula such that
δC(t) = C(t, t) < t, for every t ∈ (0, 1).
Remark 1.30 We have the following observations:
a) If we define ϕ(t) = − ln(t), then ϕ is a strict generator with ϕ[−1](t) = ϕ−1(t) = exp(−t), and using
equation (4) we have that Π2 is an Archimedean copula.
b) If we define ϕ(t) = 1 − t, then ϕ is a non-strict generator with ϕ[−1](t) = max{1 − t, 0}. Hence,
W2 is also Archimedean.
c) However, using Theorem 1.29, M2 is not Archimedean because δM2(t) = t for every t ∈ [0, 1].
6
Remark 1.31 Several families of copulas used in practice for modeling are Archimedean among
them we have:
1. Clayton Family, with generator ϕ(t) = max{0, 1
θ
(t−θ − 1), where θ ∈ [−1,∞)\{0}.
2. Ali-Mikhail-Haq (AMH), with generator ϕ(t) = ln
(
1−θ(1−t)
t
)
where θ ∈ [−1, 1).
3. Gumbel, with generator ϕ(t) = (− ln(t))θ where θ ∈ [1,∞).
4. Frank, with generator ϕ(t) = − ln
(
exp(−θt)−1
exp(−θ)−1
)
where θ ∈ (−∞,∞)\{0}.
Definition 1.32 Let Un = {(x1, y1), . . . (xn, yn)} be a random sample from a bivariate continuous
vector (X,Y), for every 1 ≤ i < j ≤ n we say that (xi, yi) and (x j, y j) are concordant if and only if
(xi − x j)(yi − y j) > 0 and they are discordant if (xi − x j)(yi − y j) < 0. The sample Kendall’s tau is
defined as
τ =
c − d
c + d
= (c − d)/
(
n
2
)
,
where c and d are the number of pairs which are concordant and discordant respectively. It can
be thought as the probability of concordance minus the probability of discordance. Therefore, the
population version of Kendall’s tau is defined by
τ = τX,Y = P[(X1 − X2)(Y1 − Y2) > 0] − P[(X1 − X2)(Y1 − Y2) < 0],
where (X1,Y1) and (X2,Y2) are independent and identically distributed random vectors with com-
mon joint distribution F.
Theorem 1.33 Let (X1,Y1) and (X2,Y2) be independent vectors of continuous random variables
with joint distributions H1 and H2, respectively, with common margins F1 of (X1 and X2) and F2 of
(Y1 and Y2). Let C1 and C2 be the respective copulas of (X1,Y1) and (X2,Y2). Let
Q = Q(C1,C2) = P[(X1 − X2)(Y1 − Y2) > 0] − P[(X1 − X2)(Y1 − Y2) < 0].
Then
Q(C1,C2) = 4
∫ ∫
[0,1]2
C2(u, v)dC1(u, v) − 1. (5)
Remark 1.34 We can see that Q(C1,C2) = Q(C2,C1). We also have that Q(M2,M2) = 1,Q(M2,Π2) =
1/3,Q(M2,W2) = 0, Q(W2,Π2) = −1/3,Q(W2,W2) = −1 and Q(Π2,Π2) = 0.
7
Lemma 1.35 If X and Y are continuous random variables with copula C. Then the population
version of Kendall’s tau is given by
τX,Y = Q(C,C) = 4
∫ ∫
[0,1]2
C(u, v)dC(u, v) − 1 = 4E(C(U,V)) − 1.
It also denoted by τC. Hence, we have that τM2 = 1, τΠ2 = 0 and τW2 = −1.7
In fact, if C1(u, v) ≤ C2(u, v) for every (u, v) ∈ [0, 1]2 then τC1
≤ τC2
, and using the Fréchet-
Hoeffding bounds we have that −1 ≤ τC ≤ 1 for every copula C.
Example 1.36 For the Clayton family Cθ with parameter θ ≥ −1, we have that τCθ
= θ/(θ + 2).
Another example is the Gumbel family with parameter θ ≥ 1, in this case, we have that τCθ
=
(θ − 1)/θ.
Example 1.37 If C is Archimedean with generator ϕ then we can find Kendall’s tau using the
formula
τC = 1 + 4
∫ 1
0
ϕ(t)
ϕ′(t)
dt.
Definition 1.38 Another measure of association is Sperman’s rho. Let (X1,Y1), (X2,Y2) and (X3,Y3)
be three independent random vectors with common continuous joint distribution F with margins
F1 and F2. The population version of Spearman’s rho is defined as proportional to the proba-
bility of concordance minus the probability of discordance of the vectors (X1,Y1) and (X2,Y3). In
fact,
vρ = ρX,Y = 3(P[(X1 − X2)(Y1 − Y3) > 0] − P[(X1 − X2)(Y1 − Y3) < 0]).
Therefore, using (5) we have that if X and Y are continuous random variables with copula C then
ρx,y = ρC = 3Q(C,Π2)
v = 12
∫ ∫
[0,1]2
uvdC(u, v) − 3
= 12
∫ ∫
[0,1]2
C(u, v)dudv − 3 (6)
Example 1.39 We have that convex combinations of copulas are copulas. Then, we can consider
the Fréchet family of copulas defined as Cα,β(u, v) = α ·M2(u, v)+ (1−α−β) ·Π2(u, v)+β ·W2(u, v)
for every α, β ≥ 0 such that α + β ≤ 1 and for every (u, v) ∈ [0, 1]2. Using the previous results, we
have that Q(Cα,β) = αQ(M2,Π2) + (1 − α − β)Q(Π2,Π2) + βQ(W2,Π2). So,
ρCα,β
= α − β.
8
We also observe that from the previous observations we have that
−1 ≤ ρC ≤ 1 for every copula C.
Remark 1.40 If U and V are uniform (0, 1) random variables with copula C, then
ρC = 12
∫ ∫
[0,1]2
uvdC(u, v) − 3
= 12E(UV) − 3
=
E(UV) − 1/4
1/12
=
E(UV) − E(U)E(V)
√
Var(U)
√
Var(V)
.
Here, we used that E(U) = 1/4 and Var(U) = 1/12. Hence, Sperman’s rho for a pair of continuous
random variables X and Y is identical to Pearson’s correlation coefficient for the random variables
U = F1(X) and V = F2(Y).
Lemma 1.41 If X and Y are continuous random variables, then
−1 ≤ 3τ − 2ρ ≤ 1.
1 + ρ
2
≥
(
1 + τ
2
)2
and
1 − ρ
2
≥
(
1 − τ
2
)2
.
For τ ≥ 0
3τ − 1
2
≤ ρ ≤ 1 + 2τ − τ2
2
,
and if τ ≤ 0
τ2 + 2τ − 1
2
≤ ρ ≤ 1 + 3τ
2
.
Definition 1.42 Let X and Y be random variables. Then X and Y are positively quadrant depen-
dent if for every (x, y) ∈ RI 2
P(X ≤ x,Y ≤ y) ≥ P(X ≤ x)P(Y ≤ y). (7)
Equivalently,
P(X > x,Y > y) ≥ P(X > x)P(Y > y).
If equation (7) holds we will write PQD(X,Y). In terms of copulas (7) can be written as
C(u, v) ≥ uv = Π2(u, v) for every (u, v) ∈ [0, 1]2.
9
Negative quadrant dependence is defined analogously, by reversing the inequalities. If PQD(X,Y)
then
3τX,Y ≥ ρX,Y ≥ 0.
Definition 1.43 Let X, Y be two random variables, we will say that Y is left tail decreasing in X,
denoted LTD(Y |X) if and only if P(Y ≤ y|X ≤ x) is a decreasing function of x for all y, equivalently,
if and only if C(u, v)/u is decreasing in u or if and only if ∂C(u, v)/∂u ≤ C(u, v)/u for almost every
u.
We will say that Y is right tail increasing in X, denoted RTI(Y |X) if and only if P(Y > y|X > x)
is a increasing function of x for all y or equivalently, if and only if 1 − u − v + C(u, v)/(1 − u) is
decreasing in u or if and only if ∂C(u, v)/∂u ≥ [v −C(u, v)]/(1 − u) for almost every u.
LTD(X|Y) RTI(X|Y) are defined by interchanging X and Y and we have that tail monotonicity
implies PQD.
We will use later on the Spearman’s rho to define a methodology to establish the order of the
sample copula in the two-dimensional case.
10
2 Comparison between the empirical copula and the sample d-
copula of order m
The d-sample copula of order m, Cm
n , is a d-copula which is a sample estimator of C(m) the checker-
board approximation of order m of a given d-copula C, see [36] and [9], as we will see the estimator
Cm
n approaches C(m) as the sample size n increases. If m is relatively large, C(m) is a very good ap-
proximation of C. Hence, Cm
n can be thought as a quasi-nonparametric method to estimate the real
d-copula C, and it becomes a nonparametric estimator when we choose the order m.
In this Chapter we make an extensive comparison of the supremum distance between the empirical
copula Cn and the real copula C, and the supremum distance between the real copula and the
sample copula of order m, Cm
n which is simply the multilinear interpolation used in the proof of
Sklar’s Theorem, based on a sample of size n and a regular partition of order m in md d-boxes
of Lebesgue measure 1/md of [0, 1]d. The same partition is used to define C(m) the checkerboard
approximation. We used different samples sizes n, and we consider a large class of frequently used
families of copulas, and some interesting families of singular copulas.
We simulated a large number of samples of each copula with sample sizes n = 20, n = 30 and n =
50, and we obtain the basic statistics of the supremum distance when we vary m from 2 up to n. We
observed that always there exist values of m, in general far smaller that n, such that the supremum
distance between the sample d-copula of order m and the real copula C gives better approximations
than using the supremum distance between the empirical copula and the real copula. We also
prove a Glivenko-Cantelli’s Theorem for the sample copula, and we provide a method to estimate
the value of m such that Cn
m the sample d-copula of order m is a good approximation of the real
d-copula C based on the simulations.
In the last section we give some remarks and observations which include an important comment on
why the case m = n is not a good option. We also see that we can easily simulate from the sample
copula Cm
n , and that these simulated samples are quite similar to the original sample. On the other
hand, we give strong evidence that it is possible to obtain a Glivenko-Cantelli’s Theorem for the
total variation distance for the checkerboard approximation C(m) and the sample copula Cm
n .
2.1 Definitions and results about the empirical copula
We start this section by recalling the principal results about the empirical copulas.
Definition 2.1 (Rank Function) Let X1, · · · , Xn be a random sample of size n of a continuous
random variables X and let X(1), · · · , X(n) be their order statistics. The rank function r : {1, . . . , n}×
11
RI n → {1, . . . , n} is defined by
r( j, X1, · · · , Xn) = k, if and only if X j = X(k) where j, k ∈ {1, . . . , n}.
Definition 2.2 (Modified Sample or Pseudosample) Let X
1
, · · · , X
n
be a random sample of size
n of a continuous random vector X of dimension d, where X
i
= (Xi,1, . . . , Xi,d) ∈ RI d, for every
i = 1, . . . , n. Let i ∈ In, the i-th modified sample Y
i
= (Yi,1, · · · ,Yi,d), is defined by
Yi, j =
1
n
r(i, X1, j, . . . , Xn, j) for every j ∈ Id.
Here we observe that the modified sample {Y
1
, . . . ,Y
n
} is always a subset of Id
Definition 2.3 (Empirical Copula) Let X
1
, · · · , X
n
be a random sample of size n of a random
vector X of dimension d, with continuous joint distribution H, where X
i
= (Xi,1, · · · , Xi,d) ∈ RI d, for
every i = 1, . . . , n. Let Y
1
, · · · ,Y
n
be the corresponding modified sample. We define the empirical
copula denoted by Cn : Id → I by
Cn(u1, . . . , ud) =
1
n
n
∑
i=1
1{Yi,1≤u1,...,Yi,d≤ud}(u1, . . . , ud) for every (u1, . . . , ud) ∈ Id. (8)
Remark 2.4 The empirical copula Cn is an approximation of the real copula C. The empirical
copula Cn given in equation (8) has jumps of magnitude 1/n at each Y
i
of the modified sample
for every i ∈ In almost surely. Hence, Cn is not continuous, and therefore Cn is not a d-copula.
However, Cn is a d-subcopula if we restrict the domain to be T d, where T = {0, 1/n, 2/n, . . . (n −
1)/n, 1}. This follows easily by observing that from the continuity of the joint distribution function
H of the random vector X, the ranks in each coordinate vary from one to n. Using Sklar’s Theorem
for the continuous joint distribution H in Definition 2.3 there exists a unique d-copula C such that
equation (3) holds.
If we are sampling from a d-copula C instead of a joint distribution function H the Definition 2.3
of the empirical copula still holds using modified samples.
A very important result about empirical copulas is the Glivenko-Cantelli’s Theorem which states
that the empirical copula Cn approaches the real copula C in supremum norm almost surely.
Theorem 2.5 (Glivenko-Cantelli) Let Cn the empirical copula constructed from a sample of size
n of a continuous joint distribution H with d-copula C, or from a d-copula C, as in Remark (2.4).
Then
lim
n→∞
sup
(u1,u2,...,ud)∈Id
|Cn(u1, u2, . . . , ud) −C(u1, u2, . . . , ud)| = 0 almost surely. (9)
12
It is quite important to observe here, that the empirical copula has been used extensively in statisti-
cal applications to model multivariate data, see for example [4], [5], [6], [7], [10], [12], [14], [16],
[18], [19], [20], [22], [23], [28], [30], [31], [39], [42], [45] and [48], just to cite some of them.
However, as observed above the empirical d-copula is not a d-copula. In order to correct this pro-
blem some authors have proposed some modifications of the empirical copulas such as the linear
B-spline copulas, see [43] and [17]. We will see later on that this approximation corresponds to our
sample d-copula of order m = n, but we will also see that, in many instances, this approximation of
the real copula does not improve the approximation given by the empirical copula. Another well
known approximation for a copula is the Bernstein copula, which is based on polynomial approx-
imations of a d-copula, see for example [41] or [36]. We will see that our proposal is far easier to
implement specially in higher dimensions and even with large sample sizes.
The empirical d-copula , based on the theory of empirical processes includes two important results
which justify its use: The first one is the Glivenko-Cantelli Theorem, stated above, which states
that for n large enough it approximates the real copula almost surely, and the second one is the
asymptotic theory which states that the normalized process converges to a Gaussian process with
a given covariance structure, see for example [8], [13] and [44]. As we will see in this paper, using
the checkerboard approximation of order m, see [36], which is a very good approximation of a
d-copula C, that when m increases, approaches rapidly C, we can obtain the Glivenko-Cantelli’s
result for the sample d-copula of order m. See also, [9] and [35].
First, we observe that in dimension one, if we have a univariate distribution function F and a
random sample X = {X1, X2, . . . , Xn} from F of size n. We know that the empirical distribution
function is defined by
Fn(x) = (1/n)
n
∑
i=1
1{Xi≤x} for every x ∈ RI .
If F is continuous then with probability one Fn has jumps of magnitude 1/n at each point Xi of the
sample. If we let F0 denote the distribution function of the constant random variable X0 = 0, and
taking order statistics we can assume that the sample satisfies −∞ < X1 < X2 < · · · < Xn < ∞, then
Fn(x) =
∑n
i=1(1/n)F0(x − Xk) for every x ∈ RI . So, if we assume that F is continuous, it is easy to
see that
supx∈RI |Fn(x) − F(x)| = max1≤k≤n (max (|Fn(Xk−) − F(Xk)|, |Fn(Xk) − F(Xk)|))
= max1≤k≤n
(
max
(∣
∣
∣
k−1
n
− F(Xk)
∣
∣
∣ ,
∣
∣
∣
k
n
− F(Xk)
∣
∣
∣
))
.
(10)
On the other hand, we have that for every k ∈ In, min(max(|(k − 1)/n − F(Xk)| , |k/n − F(xk)|))
is attained when F(Xk) = (2k − 1)/2n, and in this case min(|(k − 1)/n − F(Xk)| , |k/n − F(xk)|) =
13
max(|(k−1)/n−F(Xk)| , |k/n−F(xk)|) = 1/2n for every k ∈ In. So, if we define Xk = F−1((2k−1)/2n)
for every k ∈ In, we have, using equation (10), that
sup
x∈RI
|Fn(x) − F(x)| = 1
2n
. (11)
Therefore, we have proved the following Lemma, which we could not find a reference for it
Lemma 2.6 Let F be a univariate continuous distribution function and let X = {X1, X2, . . . , Xn} be
a random sample of size n ≥ 1 from F. Then
sup
x∈RI
|Fn(x) − F(x)| ≥ 1
2n
a.s.[PF],
where PF is the probability measure induced by F on RI .
For an upper bound on the tail probabilities we have the Dvoretzky-Kiefer-Wolfowitz inequality,
improved by Massart, see [11] and [34], which states that for every ǫ > 0 and for every n ≥ 1
P
(
sup
x∈RI
|Fn(x) − F(x)| > ǫ
)
≤ 2e−2nǫ2
. (12)
If we take ǫ = 1/(2n) in (12) we observe that that the minimum of the right hand side is attained at
n = 1 where it takes the value 2 exp(−1/2) = 1.2103 > 1 which agrees with Lemma 2.6
Let us return to the case d ≥ 2, let H be a continuous joint distribution d-dimensional and let C the
unique d-copula given in equation (3) of Sklar’sTheorem. Let X
1
, · · · , X
n
be a random sample of
size n of a random vector X of dimension d, with continuous joint distribution H, and let Y
1
, · · · ,Y
n
be the corresponding modified sample. Define the empirical copula Cn as in equation (8), then as
observed in Remark 2.4 we have that Cn : T d → [0, 1], where T = {0, 1/n, . . . , (n − 1)/n, 1} is a
d-subcopula, but not a d-copula when defined on Id. If 0 < ǫ < 1/n, since C is a d-copula then
C(ǫ, 1, . . . , 1) = ǫ. However, Cn(ǫ, 1, . . . , 1) = 0, because ǫ < 1/n. So, letting ǫ approach 1/n from
the left we have that
lim
ǫ↑(1/n)
|C(ǫ, 1, . . . , 1) −Cn(ǫ, 1, . . . , 1)| = 1/n.
Therefore we have proved the following:
Lemma 2.7 Let H be a continuous joint distribution d-dimensional for d ≥ 2, let C the unique
d-copula given in equation (3) of Sklar’sTheorem. Let X
1
, · · · , X
n
be a random sample of size n of
14
a random vector X of dimension d, with continuous joint distribution H, and let Y
1
, · · · ,Y
n
be the
corresponding modified sample. Define the empirical copula Cn as in equation (8). Then
sup
(x1,x2,...,xd)∈Id
|C(x1, . . . , xd) −Cn(x1, . . . , xd)| ≥ 1
n
a.s.[PC], (13)
where PC is the probability function induced by the copula C on RI d.
2.2 Sample d-copula of order m
We start this section by defining the concept of generalized transformation matrix given in [15]
and extended in [46] in the construction of fractal copulas.
Definition 2.8 Let In = {1, 2, · · · , n}, with n ≥ 1. For dimension d ≥ 2, let m ∈ N, we define
Id
m = ×d
i=1
Im. Let τ a probability measure in (Id
m, 2
Id
m), τ is known as a generalized transformation
matrix if for all j ∈ {1, · · · , d} and for all k ∈ {1, · · · ,m}
∑
i∈Id
m,i j=k
τ(i) > 0
where i = (i1, · · · , i j−1, i j = k, i j+1, · · · , id) ∈ Id
m. τ can be thought as a d-dimensional matrix τ,
considering
τ(i) = τi1,···,id if i = (i1, · · · , id) ∈ Id
m.
Example 2.9 Let d = 2, m = 3. Then
I2
3 = {1, 2, 3}2.
We define two probability measures τ1 and τ2 in (I2, 2I2
) given by the following matrices
τ1 =










1/6 0 0
0 1/3 0
0 0 1/2










and τ2 =










1/6 0 1/6
1/6 0 1/6
2/6 0 0










.
We can see by adding the elements in each row and column that τ1 is a transformation matrix, but
τ2 is not since the sum of the entries in the second column equals zero.
Definition 2.10 Let τ = (τi, j)i, j∈{1,···,m} be a generalized transformation matrix where d = 2. Define
{q1,0, q1,1, · · · , q1,m} and {q2,0, q2,1, · · · , q2,m} two partitions of [0, 1], such that q1,0 = q2,0 = 0 and for
i, j ∈ Im we have that
q1,i =
i
∑
i′=1
∑
j∈Im
τi′, j and q2, j =
j
∑
j′=1
∑
i∈Im
τi, j′ ,
15
we also define the partition induced by τ on I2 by
Qm
i, j =
〈
q1,i−1, q1,i
] ×
〈
q2, j−1, q2, j
]
for every (i, j) ∈ Im × Im,
where the 〈 notation indicates that the left end of the interval is closed if i = 1 or j = 1, and open
in any other case. Let Π2 be the product 2-copula, we define the τ(Π2) transformation by
τ(Π2)(u, v) =
∑
i′<i, j′< j
τi′, j′ +
u − q1,i−1
q1,i − q1,i−1
∑
j′< j
τi, j′ +
v − q2, j−1
q2, j − q2, j−i
∑
i′<i
τi′, j
+τi, jΠ
2
(
u − q1,i−1
q1,i − q1,i−1
,
v − q2, j−1
q2, j − q2, j−1
)
(14)
with u, v ∈ Qm
i, j
for every i, j ∈ Im.
It is easy to see that τ(Π2) is always a 2-copula. Evenmore, it is not difficult to see that equa-
tion (14) coincides exactly with equation (2.3.2) in Lemma 2.3.5 in the proof of Sklar’s Theorem
in [37], where a1 = q1,i−1, a2 = q1,i, b1 = q2, j−1, b2 = q2, j, λ1 = (u − q1,i−1)/(q1,i − q1,i−1),
µ1 = (v − q2, j−1)/(q2, j − q2, j−1), C′′(a1, b1) =
∑
i′<i, j′< j τi′, j′ , C′′(a1, b2) =
∑
i′<i, j′≤ j τi′, j′ , C′′(a2, b1) =
∑
i′≤i, j′< j τi′, j′ and C′′(a2, b2) =
∑
i′≤i, j′≤ j τi′, j′ . Hence equation (14) is simply a bilinear interpolation.
This definition can be extended to dimension d > 2 using the product d-copula Πd with a d-linear
interpolation.
Definition 2.11 Let m ≥ 2 and let τ = (τi1,···,id )(i1,···,id)∈Id
m
be a generalized transformation matrix.
We define q1,0 = q2,0 = · · · = qd,0 = 0, and for every j ∈ {1, · · · , d} and for every k ∈ {1, · · · ,m}
q j,k =
k
∑
i j=1
m
∑
i1=1
· · ·
m
∑
i j−1=1
m
∑
i j+1=1
· · ·
m
∑
id=1
ti1,···,i j−1,i j,i j+1,···,id .
Then 0 = q j,0 < q j,1 < · · · < q j,m−1 < q j,m = 1 is a partition of the [0, 1] interval, induced for the
matrix τ in the j-coordinate. For every i = (i1, . . . , id) ∈ Id
m we define
Qm
i = 〈q1,(i1−1), q1,i1] × 〈q2,(i2−1), q2,i2] × · · · × 〈qd,(id−1), qd,id ]. (15)
Then the family (Qm
i
)i∈Id
m
is a partition of Id.
Now we can give the definition of the sample copula of order m, based on a random sample of size
n, where 2 ≤ m ≤ n, coming from a continuous joint d-distribution function H or a d-copula C.
16
Definition 2.12 (Sample Copula of order m) Let 2 ≤ m ≤ n and let X
1
, . . . , X
n
be a random
sample of size n of a random vector X of dimension d, with continuous joint distribution H or
d-copula C, where X
i
= (Xi,1, · · · , Xi,d) ∈ RI d, for every i = 1, . . . , n. Let Un = {Y 1
, . . . ,Y
n
} be the
corresponding modified sample.
Define the uniform partition of size m of Id, where for every i = (i1, . . . , id) ∈ Id
m
Rm
i =
〈
i1 − 1
m
,
i1
m
]
× · · · ×
〈
id − 1
m
,
id
m
]
. (16)
Define
s
n,(m)
i1,...,id
=
card(Rm
i
∩ Un)
n
(17)
where card(·) denotes the cardinality of a set. Let
S n
m = (s
n,(m)
i1,...,id
)(i1,...,id)∈Id
m
, (18)
then S n
m is always a d-dimensional generalized transformation matrix. Let (Qm
i
)i∈Id
m
be the partition
of Id induced by the generalized transformation matrix S n
m given in equation (15). Using the
partition (Qm
i
)i∈Id
m
, we define the sample d-copula of order m by
Cn
m(u1, . . . , ud) = S n
m(Πd)(u1, . . . , ud), (19)
as in the generalization of equation (14), where Πd is the product copula in [0, 1]d.
To clarify this definition we give a simple example
Example 2.13 We took a sample of size n = 4 from a bivariate normal distribution with mean
µ = (0, 0) and correlation coefficient ρ = 0.5. The observations were X
1
= (0.662, 0.895), X
2
=
(−1.352,−0.174), X
3
= (1.304,−0.682), and X
4
= (0.651, 0.137), the corresponding modified
sample is U4 = {Y 1
= (3/4, 1), Y
2
= (1/4, 2/4), Y
3
= (1, 1/4, ), Y
4
= (2/4, 3/4)}.
First, let m = 2, then using equation (16) we have that
R2
1,1 ∩ U4 = {Y 2
}, R2
1,2 ∩ U4 = {Y 4
}, R2
2,1 ∩ U4 = {Y 3
}, and R2
2,2 ∩ U4 = {Y 1
}.
Then s
4,(2)
i, j
= 1/4 for every i, j ∈ I2, that is,
S 4
2 =
(
1/4 1/4
1/4 1/4
)
.
17
Therefore, S 4
2
is clearly a transformation matrix, which induces the partitions q1,0 = q2,0 = 0 <
q1,1 = q2,1 = 1/2 < q1,2 = q2,2 = 1 on the first and second coordinates, see Definition 2.10. Observe
that the partition (Q2
i, j)i, j∈I2
induced by the matrix S 4
2
, given in Definition 2.10 coincides with the
uniform partition of size 2 (R2
i, j)i, j∈I2
. So, using equation (14), it is easy to see that C4
2
= S 4
2
(Π2) is
the copula which has joint density c4
2
(u, v) = 1 for every (u, v) ∈ I2. Observe that in all four cases
c4
2
(u, v) = s
4,(2)
i, j
/λ2(Q2
i, j), where λ2 is the Lebesgue measure in RI 2.
Second, let us assume that m = 3 and let us take {R3
i, j
}i, j∈I3
the uniform partition of size 3 of I2.
Then we have that
R3
1,2 ∩ U4 = {Y 2
}, R3
2,3 ∩ U4 = {Y 4
}, R3
3,1 ∩ U4 = {Y 3
}, and R3
3,3 ∩ U4 = {Y 1
},
and for the remaining boxes R3
i, j
∩ U4 = ∅. Then
s
4,(3)
i, j
=
{
1/4 if (i, j) ∈ {(1, 2), (2, 3), (3, 1), (3, 3)}
0 if (i, j) ∈ ((I3 × I3)\{(1, 2), (2, 3), (3, 1), (3, 3)}).
Hence,
S 4
3 =










0 1/4 0
0 0 1/4
1/4 0 1/4










,
which is clearly a transformation matrix. The partitions in [0, 1] induced by the matrix S 4
3
are
q1,0 = q2,0 = 0 < q1,1 = q2,1 = 1/4 < q1,2 = q2,2 = 2/4 < q1,3 = q2,3 = 1. Define as Definition 2.10
Q3
i, j = 〈q1,i−1, q1,i] × 〈q2, j−1, q2, j] for every i, j ∈ I3.
So, using Definition 2.10 again, it is easy to see that the sample copula C4
3
= S 4
3
(Π2) of order 3,
has a joint density c4
3
given by
c4
3(u, v) =





















4 if (u, v) ∈ Q3
1,2
2 if (u, v) ∈ Q3
2,3
2 if (u, v) ∈ Q3
3,1
1 if (u, v) ∈ Q3
3,3
0 if (u, v) ∈ I2\(Q3
1,2
∪ Q3
2,3
∪ Q3
1,2
∪ Q3
1,2
)
Observe that again, in all cases c4
3
(u, v) = s
4,(3)
i, j
/λ2(Q3
i, j
).
18
Finally, assume that m = n = 4 and let us take {R4
i, j}i, j∈I4
the uniform partition of size 4 of I2. Then
we have that
R4
1,2 ∩ U4 = {Y 2
}, R4
2,3 ∩ U4 = {Y 4
}, R4
3,4 ∩ U4 = {Y 3
}, and R4
4,1 ∩ U4 = {Y 1
},
and for the remaining boxes R4
i, j ∩ U4 = ∅. Then
s
4,(4)
i, j
=
{
1/4 if (i, j) ∈ {(1, 2), (2, 3), (3, 4), (4, 1)}
0 if (i, j) ∈ ((I4 × I4)\{(1, 2), (2, 3), (3, 4), (4, 1)}).
Hence,
S 4
4 =















0 1/4 0 0
0 0 1/4 0
0 0 0 1/4
1/4 0 0 0















,
which is clearly a transformation matrix. The partitions in [0, 1] induced by the matrix S 4
4
are
q1,0 = q2,0 = 0 < q1,1 = q2,1 = 1/4 < q1,2 = q2,2 = 2/4 < q1,3 = q2,3 = 3/4 < q1,4 = q2,4 = 1. Define
as in Definition 2.10
Q4
i, j = 〈q1,i−1, q1,i] × 〈q2, j−1, q2, j] for every i, j ∈ I4,
and in this case again, the partition (Q4
i, j)i, j∈I4
of I2 coincides with the uniform partition of size 4
(R4
i, j)i, j∈I4
of order 4 of I2. So, using Definition 2.10 again, it is easy to see that the sample copula
C4
4
= S 4
4
(Π2) of order 4, has a joint density c4
4
given by
c4
4(u, v) =
{
4 if (u, v) ∈ Q4
1,2
∪ Q4
2,3
∪ Q4
3,4
∪ Q4
4,1
0 if (u, v) ∈ I2\(Q4
1,2
∪ Q4
2,3
∪ Q4
3,4
∪ Q4
4,1
)
Observe again, that c4
3
(u, v) = s
4,(4)
i, j
/λ2(Q4
i, j) for every i, j ∈ I4.
We will use the previous Example in order to see that even if Definitions 2.10, 2.11 and 2.12 seem
to be quite cumbersome, in the case of the sample copula of order m they become quite simple,
this will become apparent in the next:
Theorem 2.14 Let 2 ≤ m ≤ n and let X
1
, . . . , X
n
be a random sample of size n of a random vector
X of dimension d, with continuous joint distribution H or d-copula C, where X
i
= (Xi,1, · · · , Xi,d) ∈
RI d, for every i = 1, . . . , n. Let Un = {Y 1
, . . . ,Y
n
} be the corresponding modified sample.
19
Let 2 ≤ m ≤ n fixed and define (Rm
i
)i∈Id
m
the uniform partition of size m of Id as in equation
(16), s
n,(m)
i1,...,id
as in equation (17), the generalized transformation matrix S n
m as in equation (18), the
partition (Qm
i
)i∈Id
m
of Id induced by S n
m given in equation (15), and Cn
m the sample copula order m
as in equation (19). Then
i) For the partitions of (Qm
i
)i∈Id
m
we know that 0 = q1,0 < q1,1 < · · · < q1,m = 1, but we also have that
q j,0 = q1,0 = 0, q j,1 = q1,1, q j,2 = q1,2, . . . , q j,m = q1,m = 1 for every j ∈ {2, 3, . . . , d}, (20)
that is, in the d coordinates the partition of [0, 1] does not change. Evenmore, with probability one,
the partition 0 = q1,0 < q1,1 < · · · < q1,m = 1 only depends on n and m, and does not depend on the
sample, in fact we have that
q1, j =
1
n
·
⌊
j · n
m
⌋
for every j ∈ {0, 1, 2, . . . ,m}, (21)
where ⌊a⌋ denotes the greatest integer less than or equal to a.
ii) For every 2 ≤ m ≤ n, Cn
m is always a d-copula.
iii) Assume that m divides n, then the partition (Qm
i
)i∈Id
m
of Id induced by S n
m coincides with the
uniform partition (Rm
i
)i∈Id
m
of size m.
iv) Let λd be the Lebesgue measure on the measurable space ( RI d,B( RI d)), where B( RI d) denotes
the σ-algebra of Borel. If Cn
m is the sample copula of order m, let us denote by cn
m its joint density
function. Then
cn
m(u1, . . . , ud) = s
n,(m)
i1,...,id
/λd(Qm
i1,...,id
) for every (u1, . . . , ud) ∈ Qm
i1,...,id
and (i1, . . . , id) ∈ Id
m. (22)
Hence, the density is constant on every d-box Qm
i1,...,id
of the partition of Id induced by S n
m. Besides,
if md > n then there exists at least one d-box Qm
i1,...,id
on which the density is zero. In fact, there are
at most n d-boxes with positive density.
v) If m = n there are exactly n elements of the partition (Qm
i
)i∈Id
m
= (Rm
i
)i∈Id
m
on which the density
equals nd−1 and the remaining elements have density zero.
Proof: i) We first observe that the modified sample satisfies that in each coordinate the different
values are 1/n, 2/n, . . . , (n− 1)/n, n/n = 1, then for any 2 ≤ m ≤ n we have that equations (20) and
(21) hold. We also have that the matrix S n
m is always a generalized transformation matrix. Hence,
by the definition of the sample d-copula of order m given in equation (19) and using the results in
[15] and [46] we obtain ii).
20
Assume that m divides n, to see that iii) holds is enough to observe that we obtain always integers
in the expressions ⌊·⌋ in equation (21) and that q1, j = j/m for every j ∈ {1, 2, . . . ,m}.
iv) We know from equation (19) that the d-sample copula of order m is a multilinear function
which spreads uniformly the mass s
n,(m)
i1,...,id
over the d-box Qm
i1,...,id
for every (i1, . . . , id) ∈ Id
m. Hence
the density on each d-box Qm
i1,...,id
is a constant, the result now follows directly from evaluating the
constant in the integral expression.
v) follows directly from parts iii) and iv). 
Observe that from Theorem 2.14 the two partitions of Id, that is, the uniform partition of size m
(Rm
i
)i∈Id
m
, where 2 ≤ m ≤ n, and the partition induced by the generalized transformation matrix S n
m,
(Qm
i
)i∈Id
m
do not always coincide. Using equation (16), if we define 0 = rk,0 < 1/m = rk,1 < 2/m =
rk,2 < · · · < (m−1)/m = rk,m−1 < 1 = rk,m for every k ∈ {1, 2, . . . , d}, then (rk, j)k∈Id , j∈{0,1,...,m} provides
the partition in each coordinate induced by the uniform partition of size m. We define the distance
between (Rm
i
)i∈Id
m
and (Qm
i
)i∈Id
m
by
em((Rm
i ), (Qm
i )) = max
j∈{0,1,...,m}
|r1, j − q1, j|,
where q1, j is given in equation (21). Then em measures the “distortion” of the uniform partition
of size m, caused by the sample size n. It is easy to see that max2≤m≤n em((Rm
i
), (Qm
i
)) ≤ (n −
2)/((n − 1)n) < 1/n, that is, the maximum is attained when m = n − 1, and this maximum distance
is always smaller that 1/n. Using part iii) of Theorem 2.14, we have that if m divides n, then
em((Rm
i
), (Qm
i
)) = 0.
In Theorem 2.14, part iv) we give the joint density cn
m associated to Cn
m the sample copula of
order m, from this density the evaluation of Cn
m is absolutely trivial, and easily implemented in any
computer.
Part v) of Theorem 2.14, implies that most of the d-boxes in the partition of Id have zero den-
sity. For example if the sample size n = 100 and d = 4, we have that only one hundred out of
100, 000, 000 4-boxes have positive density. In this example is quite important to notice that the
one hundred 4-boxes with positive density include the support of the empirical copula Cn, which
is included in T d, where T = {0, 1/n, 2/n, . . . , 1} given in Definition 2.3 and Remark 2.4. Besides,
the sample copula Cn
n is a d-copula unlike the empirical copula Cn, which is only a d-subcopula.
21
Now we will see, that in some cases, Cn
m the sample copula of order m may coincide with C, the
d-copula we are sampling from, that is, sup(u1,...,ud)∈Id |Cn
m(u1, . . . , ud) −C(u1, . . . , ud)| = 0.
Lemma 2.15 Let d ≥ 2 be an integer and let n ≥ 2 be an even integer. Then there exist 2 ≤ m ≤ n,
C a d-copula and a sample of size n from C, such that
sup
(u1,...,ud)∈Id
|Cn
m(u1, . . . , ud) −C(u1, . . . , ud)| = 0. (23)
Proof: Let d ≥ 2 be an integer and let n ≥ 2 be an even integer. Let C be a d-copula with density
c given by
c(u1, . . . , ud) =
{
2d−1 if (u1, . . . , ud) ∈ [0, 1/2]d ∪ (1/2, 1]d
0 if (u1, . . . , ud) ∈ Id\([0, 1/2]d ∪ (1/2, 1]d).
Let m = 2. The uniform partition of size 2 of Id is such that R1,1,...,1 = [0, 1/2]d and R2,2,...,2 =
(1/2, 1]d accumulate all the mass of the d-copula C. Let X
1
, . . . , X
n
be a random sample of size n
from the d-copula C, and assume that exactly n/2 points of the sample fall in the d-box R1,1,...,1, then
the remaining n/2 points fall in the box R2,2,...,2 with probability one, because the density is zero
on any other d-box of the uniform partition of size 2. Let Un = {Y 1
, . . . ,Y
n
} be the corresponding
modified sample, then it is obvious that this modified sample satisfies exactly the same conditions.
Hence, using equation (17), s
n,(2)
1,1,...,1
= s
n,(2)
2,2,...,2
= 1/2 and the remaining s
n,(2)
i1,i2,...,id
= 0. Therefore,
using Theorem 2.14, part iv), the density of the sample d-copula of order 2 is given by
cn
m(u1, . . . , ud) =
{
2d−1 if (u1, . . . , ud) ∈ [0, 1/2]d ∪ (1/2, 1]d
0 if (u1, . . . , ud) ∈ Id\([0, 1/2]d ∪ (1/2, 1]d),
which is exactly the density of the d-copula C, and the result follows. 
However, observe that from Lemma 2.7, the empirical copula Cn satisfies that
sup
(u1,...,ud)∈Id
|Cn(u1, . . . , ud) −C(u1, . . . , ud)| ≥ 1/n
almost surely.
Since we are using the product copula in order to define the sample d-copula of order m, see
equation (19), we have to see what is the largest error we can incur by doing so. First we observe
that for d = 2, we have that
sup
(u,v)∈I2
|Π2(u, v) − M2(u, v)| = sup
(u,v)∈I2
|Π2(u, v) −W2(u, v)| = 1/4, (24)
22
where the supremum is attained at u = v = 1/2 in both cases, as can be easily checked. Hence,
from equation (24) we have that for every 2-copula C, using the Fréchet-Hoeffding’s bounds,
sup(u,v)∈I2 |Π2(u, v) −C(u, v)| ≤ 1/4.
For the case d ≥ 3, we have that
sup
(u1,...,ud)∈Id
|Πd(u1, . . . , ud) − Md(u1, . . . , ud)| = d − 1
dd/(d−1)
, (25)
where the supremum is attained at u1 = u2 = · · · ud = 1/d1/(d−1).
For every C1,C2 d-copulas let us define
dsup(C1,C2) = sup
(u1,u2,...,ud)∈Id
|C1(u1, u2, . . . , ud) −C2(u1, u2, . . . , ud)|. (26)
Then clearly dsup is a metric in the set of all d-copulas.
Definition 2.16 Let 2 ≤ m ≤ n and let X
1
, . . . , X
n
be a random sample of size n of a random
vector X of dimension d, with continuous joint distribution H with d-copula C, or from the d-
copula C where X
i
= (Xi,1, · · · , Xi,d) ∈ RI d, for every i = 1, . . . , n. Let Un = {Y 1
, . . . ,Y
n
} be the
corresponding modified sample.
Let Cn be the empirical copula defined in equation (8), and let Cn
m be the sample copula of order
m defined as in equation (19) of Definition 2.12. We define
dsup
n
(Cn,C) = max







sup
(i1,i2,...,id)∈In
d
|Cn(i1/n, i2/n, . . . , id/n) −C(i1/n, i2/n, . . . , id/n)| , 1
n







(27)
and
dsup
n,(m)
(Cn
m,C) = sup
(i1,i2,...,id)∈In
d
|Cn
m(i1/n, i2/n, . . . , id/n) −C(i1/n, i2/n, . . . , id/n)|. (28)
By equation (13) we now that dsup(Cn,C) ≥ 1/n almost surely, that is why the term 1/n appears
in equation (27). Hence dsup
n
(Cn,C) is never a metric. However, dsup
n,(m)
(Cn
m,C) in equation (28)
is a pseudometric in the family of all d-copulas. In order to see that this statement holds just
observe that Cn
m is always a d-copula, and for any d-copula C such that Cn
m(i1/n, i2/n, . . . , id/n) =
C(i1/n, i2/n, . . . , id/n) for every (i1, i2, . . . , id) ∈ In
d we have that dsup
n,(m)
(Cn
m,C) = 0. In particular,
if C = Cn
m we have that dsup
n,(m)
(Cn
m,C) = 0.
23
Of course we have from equations (26), (27) and (28)that
dsup(Cn,C) ≥ dsup
n
(Cn,C) and dsup(Cn
m,C) ≥ dsup
n,(m)
(Cn
m,C), (29)
for every integers 2 ≤ m ≤ n and for every d-copula C. We will show with an easy example that
the first inequality in equation (29) can be strict, but the second one is an equality.
Let d = 2, n = 2 and let C = Π2 be the product copula. Then the modified sample is U2 =
{(1/2, 2/2 = 1), (2/2 = 1, 1/2)} or U2 = {(1/2, 1/2), (1 = 2/2, 1 = 2/2)} each with probability
1/2. In the first case, the mass of the empirical copula is concentrated in the two points (1/2, 1)
and (1, 1/2), therefore, it is quite easy to see from equation (27) that dsup
2
(C2,Π
2) = max{|0 −
1/4|, 1/2} = 1/2. But, if we take any 0 < ǫ < 1 then we have that dsup(C2,Π
2) ≥ |C2(1− ǫ, 1− ǫ)−
(1−ǫ)2| = (1−ǫ)2, so if we let ǫ ↓ 0 we have that dsup(C2,Π
2) = 1 > 1/2 = dsup
2
(C2,Π
2). Observe
that in this example we obtain the upper bound, since dsup(C,D) ≤ 1 for any two distribution
functions C and D on I2. Since n = 2, then we have that m = 2 for the sample copula, and
in this case we have that the sample copula of order m = 2 gives uniform masses 1/2 to the
boxes R1,2 = [0, 1/2] × (1/2, 1] and R2,1 = (1/2, 1] × [0, 1/2], so using equation (28) we have that
dsup
2,(2)
(C2
2
,Π2) = |C2
2
(1/2, 1/2) − Π2(1/2, 1/2)| = 1/4 = dsup(C2
2
,Π2).
In the second case, the mass of the empirical copula is concentrated in the two points (1/2, 1/2) and
(1, 1), so, from equation (27) we have that dsup
2
(C2,Π
2) = max{|1/2−1/4|, 1/2} = 1/2. Using (13)
we can see that dsup(C2,Π
2) = 1/2. In this case we have that the sample copula of order m = 2
gives uniform masses 1/2 to the boxes R1,1 = [0, 1/2]2 and R2,2 = (1/2, 1]2, so using equation (28)
we have that dsup
2,(2)
(C2
2
,Π2) = |C2
2
(1/2, 1/2) − Π2(1/2, 1/2)| = 1/4 = dsup(C2
2
,Π2).
Let X
1
, . . . , X
n
be a random sample of size n from the d-copula Md. Let Un = {Y 1
, . . . ,Y
n
} be the
corresponding modified sample, then {Y
i
= (i/n, i/n, . . . , i/n)}n
i=1
almost surely. So, it is obvious
that sup(i1,i2,...,id)∈In
d |Cn(i1/n, i2/n, . . . , id/n) − Md(i1/n, i2/n, . . . , id/n)| = 0, but from equation (13)
dsup(Cn,C) = 1/n.
We finish this section by proving a Glivenko-Cantelli’s Theorem for the sample d-copula of order
m.
Theorem 2.17 Let C be a d-copula, let m ≥ 2 and let n be a multiple of m, let Cn
m be the sample
copula built from a modified sample of size n from C. Then
lim
m−→∞
sup
(u,v)∈I2
∣
∣
∣Cn
m(u, v) −C(u, v)
∣
∣
∣ = 0 a.s.
24
Proof: We prove the result for d = 2. Using the notation in Lemma 2.3.5 in Nelsen’s book [37]:
For (a, b) ∈ [0, 1]2 and a subcopula C′ with domain finite S 1 × S 2, let a1 and a2 be, respectively,
the greatest and least elements of S 1 that satisfy a1 ≤ a ≤ a2; and let b1 and b2 be, respectively,
the greatest and least elements of S 2 that satisfy b1 ≤ b ≤ b2. If a ∈ S 1, then a1 = a = a2, and if
b ∈ S 2, then b1 = b = b2. Let
λ1(a, b) = λ1 =
{
(a − a1)/(a2 − a1) if a1 < a2
1 if a1 = a2
and
µ1(a, b) = µ1 =
{
(b − b1)/(b2 − b1) if b1 < b2
1 if b1 = b2.
We will consider the following representation of the the sample copula Cn
m, see [26].
Cn
m(u, v) =
m
∑
j=1
m
∑
i=1
[
1( i−1
m
, i
m ]×
(
j−1
m
,
j
m
](u, v)
(
(1 − λ1(u, v))(1 − µ1(u, v))Cn
(
i − 1
m
,
j − 1
m
)
+(1 − λ1(u, v))µ(u, v)1Cn
(
i − 1
m
,
j
m
)
+ λ1(u, v)(1 − µ1(u, v))Cn
(
i
m
,
j − 1
m
)
+ λ1(u, v)µ1(u, v)Cn
(
i
m
,
j
m
))]
where Cn is the empirical copula built from the modified sample, and the checkerboard approxi-
mation C(m), see [33], given by
C(m)(u, v) =
m
∑
j=1
m
∑
i=1
[
1( i−1
m
, i
m ]×
(
j−1
m
,
j
m
](u, v)
(
(1 − λ1(u, v))(1 − µ1(u, v))C
(
i − 1
m
,
j − 1
m
)
+(1 − λ1(u, v))µ1(u, v)C
(
i − 1
m
,
j
m
)
+ λ1(u, v)(1 − µ1(u, v))C
(
i
m
,
j − 1
m
)
+ λ1(u, v)µ1(u, v)C
(
i
m
,
j
m
))]
for all (u, v) ∈ I2.
Let (u∗, v∗) ∈ I2 and m ≥ 2, where m divides n, such that (u∗, v∗) ∈ Rm
i∗ j∗ =
〈
i∗−1
m
, i∗
m
]
×
〈
j∗−1
m
,
j∗
m
]
where the notation “〈” indicates that the left end of interval is closed if i∗ = 1 or j∗ = 1 and open
in any other case, then
|Cn
m(u∗, v∗) −C(m)(u∗, v∗)| ≤ (1 − λ1(u∗, v∗))(1 − µ1(u∗, v∗))
∣
∣
∣
∣
∣
∣
Cn
(
i∗ − 1
m
,
j∗ − 1
m
)
−C
(
i∗ − 1
m
,
j∗ − 1
m
)
∣
∣
∣
∣
∣
∣
25
+(1 − λ1(u∗, v∗))µ1(u∗, v∗)
∣
∣
∣
∣
∣
∣
Cn
(
i∗ − 1
m
,
j∗
m
)
−C
(
i∗ − 1
m
,
j∗
m
)
∣
∣
∣
∣
∣
∣
+λ1(u∗, v∗)(1 − µ1(u∗, v∗))
∣
∣
∣
∣
∣
∣
Cn
(
i∗
m
,
j∗ − 1
m
)
−C
(
i∗
m
,
j∗ − 1
m
)
∣
∣
∣
∣
∣
∣
+λ1(u∗, v∗)µ1(u∗, v∗)
∣
∣
∣
∣
∣
∣
Cn
(
i∗
m
,
j∗
m
)
−C
(
i∗
m
,
j∗
m
)
∣
∣
∣
∣
∣
∣
.
Hence, using Glivenko-Cantelli’s Theorem for the empirical copula, we have that
lim
m−→∞
sup
(u,v)∈I2
∣
∣
∣Cn
m(u, v) −C(m)(u, v)
∣
∣
∣ = 0 a.s.
From [33], we have that, for every m ≥ 1,
sup
(u,v)∈I2
∣
∣
∣C(m)(u, v) −C(u, v)
∣
∣
∣ <
2
m
,
and
lim
m−→∞
sup
(u,v)∈I2
∣
∣
∣C(m)(u, v) −C(u, v)
∣
∣
∣ = 0.
From these relations we get that
lim
m−→∞
sup
(u,v)∈I2
∣
∣
∣Cn
m(u, v) −C(u, v)
∣
∣
∣ = 0 a.s.
The proof of the case d-dimensional, where d > 2, is performed similarly. 
2.3 Simulation study: dimension two
For this section we performed a large simulation study in order to compare the supremum distance
defined on equation (26), between the real copula and the empirical copula and then the supremum
distance between the real copula and the sample copula of order m. Here we used equations (27)
and (28), which are good approximations of the supremum distance (26).
In this subsection, we study twenty five different families in dimension d = 2, which include
several of the most used families of copulas such as: Ali-Mikhail-Haq, Clayton, Frank, Gumbel,
Joe, Normal, Plackett, Product and t-Student. We also include some copulas that are singular such
as M2, W2, Examples 3.3, 3.4, 3.5 and the circular distribution copula from Nelsen‘s book [37]. We
also include an example of an absolutely continuous copula whose support is [0, 1/2]2 ∪ (1/2, 1]2
26
and we call it the protocol copula. We also study increasing and decreasing transformations of
the coordinates of a copula in the Plackett and the Gumbel families. Finally we study a couple of
examples of mixtures of copulas to study the behavior of asymmetric copulas. The study consisted
in taking random samples of sizes n = 20, 30 and n = 50 with N = 2000 or N = 10000 repetitions
for the first two values of n and N = 1500 repetitions for n = 50. We obtained the sample copulas
Cn
m for every 2 ≤ m ≤ n and the empirical copula Cn and we approximate the supremum distance
using equations (27) and (28). Finally we report the mean values, minima and maxima of the N
iterations in the first figures, the straight lines correspond to the minima, mean values and maxima
of dsup
n
(Cn,C) and the other lines correspond to the same statistics for dsup
n,(m)
(Cn
m,C). In the
second figures we report for some of the cases the comparisons of the variances.
We report the values of Spearman’s ρ, which is a common concordance measure, instead of the
parameters of the families in order to make comparisons.
From now on, we will say that Cn
m is a better approximation than Cn to the real copula C for
a given value of 2 ≤ m ≤ n if the mean value of the iterations satisfy that dsup
n,(m)
(Cn
m,C) ≤
dsup
n
(Cn,C). We will make some remarks about the variances later on.
First, we start to the case of the product copula Π2, see Figure 2.11, in which we show that for
n = 20, 30 and n = 50 we have that for every 2 ≤ m ≤ n, Cn
m is a better approximation than Cn,
and the variances in all three cases are quite similar for 3 ≤ m ≤ n, even for m = 2 where the
variance difference is the greatest we also have that the difference between the mean values is quite
remarkable. However, these facts are expected since in the construction of the d-sample copula of
order m we used uniform masses.
Second, for the Ali-Mikhail-Haq family we have in Figure 2.1 the results for θ ∈ (−1, 0) and in
Figure 2.2 for θ ∈ (0, 1). As we can see for every 2 ≤ m ≤ n Cm
20
is a better approximation than C20,
and in some instances these mean values are even smaller than the minimum of dsup
m,(20)
(C20,C)
for m = 2. The behavior of this family follows since it is known that its dependence is short of
independence, because −0.27106 ≤ ρ ≤ 04784.
Third, the Clayton family is a symmetric one and its results are given in Figures 2.3 and Figure
2.4, the first one includes the results for −1 ≤ ρ < 0, we know that the limit of this family as θ
approaches −1 is W2, an when θ approaches zero its limit is the product copula Π2, and for θ close
to −1 is quite similar to the case W2. Observe that for every m ≥ 5 we have that C20
m is a better
approximation than C20. In Figure 2.4 we show the results for 0 < ρ ≤ 1, in this case we know
that whenever θ increases to ∞ the copula tends to M2. Recall that a family which has limits W2
27
and M2 it is called comprehensive, here we also observe that for every m ≥ 5 we have that C20
m is a
better approximation than C20.
Fourth, the Frank family is also a symmetric and comprehensive set of copulas when θ tends to
−∞ the copula approaches W2, and when θ tends to∞ the copula approaches M2. We observe that
for values of ρ which are not too far from zero we have that C20
m is a better approximation than C20
for several values of m, see Figure 2.5 and Figure 2.6. In fact, this holds to be true for any ρ if
m ≥ n/2 = 10.
Fifth, the Gumbel family is a symmetric family such that for θ = 1 we obtain the product copula
Π2 with ρ = 0, and when θ tends to ∞ the copula approaches M2. Again, for values of ρ close to
zero C20
m is a better approximation than C20 for every 2 ≤ m ≤ 20, and for large values of ρ it also
holds for m ≥ n/2 = 10 see Figure 2.7.
Sixth, the Joe family is also symmetric and it behaves quite similar to the Gumbel family, see
Figure 2.8.
Seventh, the normal family is symmetric, comprehensive and it is parametrized via its correlation
coefficient ρ, it can be seen that for |ρ| ≤ 0.9, C20
m is a better approximation than C20 for every
3 ≤ m ≤ n. But for |ρ| close to one it starts to behaves like W2 or M2 depending on the sign of ρ.
Eighth, the Plackett family is symmetric with parameter ρ ∈ (−1, 1), for θ near zero it behaves
like W2, for θ = 1 it is Π2, and for very large values of θ it behaves like M2, so this family is
also comprehensive. Hence, C20
m is a better approximation than C20 for several values of m if |ρ| is
relatively larger than zero and not too large, see Figure 2.10.
Ninth, the t-Student family is also symmetric, comprehensive and it is parametrized via its corre-
lation coefficient ρ, it can be seen that for |ρ| ≤ 0.95, C20
m is a better approximation than C20 for
every 2 ≤ m ≤ 20. The only difference respect to the normal case is the fact that the t-Student has
heavier tail behavior, see Figure 2.12.
Tenth, the cases of M2 and W2 are of particular importance. To begin with they are the upper
and lower bounds for every copula, besides, they represent extreme dependence, because they
are the copulas of random variables X and Y for which Y is a strictly increasing or a strictly
decreasing function of X, with ρ = 1 and ρ = −1, respectively. Besides, both copulas are singular
with supports the main diagonal and the second diagonal, respectively. From Figure 2.13 left two
graphs, we see that they have similar behavior and that for m ≥ 5 we have that C20
m is a better
28
approximation than C20. As we have seen above, several families of copulas with limit cases M2
or W2 have pretty similar behavior to the first two graphs in Figure 2.13. It is also very important
to observe that for any given sample of size n, the minimum and the maximum coincide with the
mean value, this happens because the modified samples are always the same.
Eleventh, the cases of the circular uniform distribution and Example 3.5 given in Nelsen’s book
[37] are examples of singular copulas, the second one with support on two quarters of circles of
radius one, see Figure 3.5 in [37], as we can see in the right graphs of Figure 2.13 we have that C20
m
is a better approximation than C20 for every 2 ≤ m ≤ 20.
Twelfth, the case of Example 3.3 in [37] is another case of singular copulas with given support the
segment of lines joining (0, 0) with (θ, 1), and (θ, 1) with (1, 0), for θ ∈ [0, 1]. As we can see in
Figure 2.14 the behavior of the difference between the supremum distances behaves like the case
of M2 for ρ = θ = 1 and like W2 for ρ = θ = 0.
Thirteenth, the case of Example 3.4 in Nelsen’s book [37], it is simply shuffles of M2 with support
the segment of lines joining (0, θ) with (θ, 0), and (θ, 1) with (1, θ), where θ ∈ [0, 1]. for a general
definition of shuffle see [35] or [37], for the multivariate case see [9]. In this case, see Figure 2.15,
we have that C20
m is a better approximation than C20 for 10 = n/2 ≤ m ≤ n = 20 with a similar
behavior to W2. This case is the one that presents smaller differences between the mean values for
0.1 ≤ θ ≤ 0.9.
Fourteenth, the case of what we called the protocol copula, which has incomplete support given
by [0, 1/2]2 and (1/2, 1]2 with uniform masses on each square it is quite important. We can see
from Figure 2.16 left graph, that C20
2
is a better approximation than C20, but C20
3
is not a better
approximation than C20, in fact, we can see that the behavior of the graph oscillates between
even and odd values of m this can be easily explained due to the form of its support. It is also
very important to see that the minimum of the observed supremum distances between C20
2
and the
protocol copula is zero.
Fifteenth, recall that from Theorem 2.4.4 in [37] if two random variables U and V have copula C,
then we can give just in terms of C the copulas of the pairs (U, 1−V), (1−U,V) and (1−U, 1−V), we
will denote these cases by ID, DI and DD, where I means a strictly increasing transformation and
D denotes a strictly decreasing transformation. We studied these transformation for the Gumbel
and the Plackett families, we choose these two families to obtain asymmetric copulas with the
transformations ID and DI. The result for the Gumbel family are shown in Figure 2.16 last three
graphs, Figure 2.17 and Figure 2.18. For the Plackett family we refer to Figure 2.19, Figure 2.20
29
and Figure 2.21. We can observe that the values of the parameters that we used were the same for
each family, and that the graphs look very much alike in each case.
Sixteenth, the last three cases are mixtures; First of Gumbel, GumbelID (GID), and GumbelDI
(GDI), second GID, Frank and Joe, in both cases we obtain highly asymmetric copulas; and third
a mixture of M2 and W2 which is singular. The results are shown in Figure 2.22, Figure 2.23 and
Figure 2.24 and the comments are quite similar to some of the previous cases. For the PlackettID
we will use the abbreviation (PID), for PlackettDI (PDI) and for PlackettDD (PDD).
Summing up, the results for all the families of copulas absolutely continuous with complete support
are quite similar as a function of the Spearman’s ρ. However, for the families of singular copulas
the results are sometimes comparable to those of M2 or W2, and they depend strongly on their
supports. When the copula is absolutely continuous, but it does not have complete support, the
behavior may be alittle erratic as shown by the protocol copula in Figure 2.16.
We can observe that when m = n, the sample copula Cn
m is at least as good an approximation to
the real copula C as the empirical copula Cn, and in many cases a better approximation, when |ρ| is
close to one. In fact, in many cases the best approximation is given with values of m quite smaller
than n, and this value of m depends strongly on the value of Spearman’s ρ.
30
5 10 15 20
m value (minimun sample mean copula for m = 2)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35 Supremum Distance AMH(-0.99,d=2)n20iters10000
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
5 10 15 20
m value (minimun sample mean copula for m = 2)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
Supremum Distance AMH(-0.7,d=2)n20iters10000
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
5 10 15 20
m value (minimun sample mean copula for m = 2)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
Supremum Distance AMH(-0.5,d=2)n20iters10000
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
5 10 15 20
m value (minimun mean sample copula for m = 2)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
Supremum Distance AMH(-0.1,d=2)n20iters1000
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
Figure 2.1: Ali-Mikhail-Haq d = 2 for ρ = −0.2688,−0.2004,−0.1489 and −0.0325 with n = 20
5 10 15 20
m value (minimun sample mean copula for m = 2)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
Supremum Distance AMH(0.1,d=2)n20iters10000
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
5 10 15 20
m value (minimun sample mean copula for m = 2)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
Supremum Distance AMH(0.5,d=2)n20iters10000
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
5 10 15 20
m value (minimun sample mean copula for m = 2)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
Supremum Distance AMH(0.9,d=2)n20iters10000
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
5 10 15 20
m value (minimun sample mean copula for m = 3)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
Supremum Distance AMH(0.99,d=2)n20iters10000
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
Figure 2.2: Ali-Mikhail-Haq d = 2 for ρ = 0.0342, 0.1924, 0.4070 and 0.4706 with n = 20
5 10 15 20
m value (minimun sample mean copula for m = 20)
0.00
0.05
0.10
0.15
0.20
Supremum Distance Clayton(-0.99,d=2)n20iters10000
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
5 10 15 20
m value (minimun mean sample copula for m = 7)
0.00
0.05
0.10
0.15
0.20
0.25
Supremum Distance Clayton(-0.9,d=2)n20iters2000
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
5 10 15 20
m value (minimun mean sample copula for m = 3)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
Supremum Distance Clayton(-0.5,d=2)n20iters2000
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
5 10 15 20
m value (minimun mean sample copula for m = 2)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
Supremum Distance Clayton(-0.1,d=2)n20iters2000
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
Figure 2.3: Clayton d = 2 for ρ = −0.99,−0.8978,−0.4670 and −0.0787 with n = 20
5 10 15 20
m value (minimun mean sample copula for m = 2)
0.00
0.05
0.10
0.15
0.20
0.25
Supremum Distance Clayton(0.5,d=2)n20iters2000
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
5 10 15 20
m value (minimun mean sample copula for m = 5)
0.00
0.05
0.10
0.15
0.20
0.25
Supremum Distance Clayton(5,d=2)n20iters2000
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
5 10 15 20
m value (minimun mean sample copula for m = 6)
0.00
0.05
0.10
0.15
0.20
0.25
Supremum Distance Clayton(10,d=2)n20iters10000
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
5 10 15 20
m value (minimun sample mean copula for m = 10)
0.00
0.05
0.10
0.15
0.20
Supremum Distance Clayton(25,d=2)n20iters10000
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
Figure 2.4: Clayton d = 2 for ρ = 0.2955, 0.8848, 0.9582 and 0.9915 with n = 20
31
5 10 15 20
m value (minimun mean sample copula for m = 5)
0.00
0.05
0.10
0.15
0.20
Supremum Distance Frank(-20,d=2)n20iters2000
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
5 10 15 20
m value (minimun mean sample copula for m = 4)
0.00
0.05
0.10
0.15
0.20
0.25
Supremum Distance Frank(-10,d=2)n20iters2000
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
5 10 15 20
m value (minimun mean sample copula for m = 3)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
Supremum Distance Frank(-5,d=2)n20iters2000
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
5 10 15 20
m value (minimun mean sample copula for m = 2)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
Supremum Distance Frank(-1,d=2)n20iters2000
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
Figure 2.5: Frank d = 2 for ρ = −0.9578,−0.8602,−0.6434 and −0.1644 with n = 20
5 10 15 20
m value (minimun sample mean copula for m = 2)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
Supremum Distance Frank(2,d=2)n20iters10000
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
5 10 15 20
m value (minimun mean sample copula for m = 3)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
Supremum Distance Frank(7,d=2)n20iters2000
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
5 10 15 20
m value (minimun mean sample copula for m = 5)
0.00
0.05
0.10
0.15
0.20
0.25
Supremum Distance Frank(15,d=2)n20iters2000
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
5 10 15 20
m value (minimun sample mean copula for m = 4)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
Supremum Distance Frank(20,d=2)n20iters10000
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
Figure 2.6: Frank d = 2 for ρ = 0.3168, 0.7630, 0.9293 and 0.9578 with n = 20
5 10 15 20
m value (minimun mean sample copula for m = 3)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
Supremum Distance Gumbel(2,d=2)n20iters2000
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
5 10 15 20
m value (minimun mean sample copula for m = 10)
0.00
0.05
0.10
0.15
0.20
Supremum Distance Gumbel(15,d=2)n20iters2000
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
5 10 15 20
m value (minimun sample mean copula for m = 10)
0.00
0.05
0.10
0.15
0.20
Supremum Distance Gumbel(20,d=2)n20iters10000
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
5 10 15 20
m value (minimun mean sample copula for m = 10)
0.00
0.05
0.10
0.15
0.20
Supremum Distance Gumbel(50,d=2)n20iters2000
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
Figure 2.7: Gumbel d = 2 for ρ = 0.6828, 0.9935, 0.9963 and 0.999 with n = 20
5 10 15 20
m value (minimun sample mean copula for m = 7)
0.00
0.05
0.10
0.15
0.20
0.25
Supremum Distance Joe(15,d=2)n20iters10000
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
5 10 15 20
m value (minimun sample mean copula for m = 10)
0.00
0.05
0.10
0.15
0.20
Supremum Distance Joe(25,d=2)n20iters10000
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
5 10 15 20
m value (minimun sample mean copula for m = 10)
0.00
0.05
0.10
0.15
0.20
Supremum Distance Joe(50,d=2)n20iters10000
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
5 10 15 20
m value (minimun sample mean copula for m = 10)
0.00
0.05
0.10
0.15
0.20
Supremum Distance Joe(75,d=2)n20iters10000
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
Figure 2.8: Joe d = 2 for ρ = .9766, .9908, .9975 and .9988 with n = 20
32
5 10 15 20
m value (minimun mean sample copula for m = 7)
0.00
0.05
0.10
0.15
0.20
Supremum Distance Normal(-0.99,d=2)n20iters2000
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
5 10 15 20
m value (minimun sample mean copula for m = 2)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
Supremum Distance Normal(-0.5,d=2)n20iters10000
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
5 10 15 20
m value (minimun sample mean copula for m = 2)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
Supremum Distance Normal(0.1,d=2)n20iters10000
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
5 10 15 20
m value (minimun sample mean copula for m = 3)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
Supremum Distance Normal(0.7,d=2)n20iters10000
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
Figure 2.9: Normal d = 2 for ρ = −0.9889,−0.4825, 0.0955 and 0.6829 with n = 20
5 10 15 20
m value (minimun sample mean copula for m = 8)
0.00
0.05
0.10
0.15
0.20
Supremum Distance Plackett(0.001,d=2)n20iters10000
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
5 10 15 20
m value (minimun sample mean copula for m = 2)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
Supremum Distance Plackett(0.5,d=2)n20iters10000
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
5 10 15 20
m value (minimun mean sample copula for m = 3)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
Supremum Distance Plackett(10,d=2)n20iters2000
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
5 10 15 20
m value (minimun sample mean copula for m = 10)
0.00
0.05
0.10
0.15
0.20
Supremum Distance Plackett(1000,d=2)n20iters10000
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
Figure 2.10: Plackett d = 2 for ρ = −0.9881,−0.2274, 0.6536 and 0.9881 with n = 20
5 10 15 20
m value (minimun sample mean copula for m = 2)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
Supremum Distance Producto(d=2)n20iters10000
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
5 10 15 20 25 30
m value (minimun sample mean copula for m = 2)
0.00
0.05
0.10
0.15
0.20
0.25
Supremum Distance Producto(d=2)n30iters10000
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
10 20 30 40 50
m value (minimun sample mean copula for m = 2)
0.00
0.05
0.10
0.15
0.20
Supremum Distance Producto(d=2)n50iters1500
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
Figure 2.11: Product d = 2 with ρ = 0 for n = 20, 30 and 50
5 10 15 20
m value (minimun mean sample copula for m = 2)
0.00
0.05
0.10
0.15
0.20
0.25
Supremum Distance Tstudent(-0.6,d=2)n20iters2000
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
5 10 15 20
m value (minimun mean sample copula for m = 2)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
Supremum Distance Tstudent(-0.3,d=2)n20iters2000
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
5 10 15 20
m value (minimun mean sample copula for m = 2)
0.00
0.05
0.10
0.15
0.20
0.25
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Supremum Distance Tstudent(0.5,d=2)n20iters2000
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empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
5 10 15 20
m value (minimun mean sample copula for m = 7)
0.00
0.05
0.10
0.15
0.20
Supremum Distance Tstudent(0.99,d=2)n20iters10000
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
Figure 2.12: t-Student d = 2 for ρ = −0.5819,−0.2875, 0.4825 and 0.9889 with n = 20
33
5 10 15 20
m value (minimun sample mean copula for m = 20)
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
Supremum Distance M(d=2)n20iters2
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
5 10 15 20
m value (minimun sample mean copula for m = 20)
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
Supremum Distance W(d=2)n20iters2
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
5 10 15 20
m value (minimun sample mean copula for m = 2)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35 Supremum Distance Circular(d=2)n20iters10000
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
5 10 15 20
m value (minimun sample mean copula for m = 2)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
Supremum Distance Example3.5(d=2)n20iters10000
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
Figure 2.13: M2ρ = 1, W2ρ = −1, Circular ρ = 0 and Examp. 3.5 of Nelsen ρ.286 with n = 20
5 10 15 20
m value (minimun mean sample copula for m = 11)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
Supremum Distance Example3.3(0.05,d=2)n20iters10000
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
5 10 15 20
m value (minimun mean sample copula for m = 6)
0.0
0.1
0.2
0.3
0.4
Supremum Distance Example3.3(0.3,d=2)n20iters10000
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
5 10 15 20
m value (minimun mean sample copula for m = 7)
0.0
0.1
0.2
0.3
0.4
Supremum Distance Example3.3(0.6,d=2)n20iters10000
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
5 10 15 20
m value (minimun mean sample copula for m = 10)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
Supremum Distance Example3.3(0.9,d=2)n20iters10000
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
Figure 2.14: Example 3.3 of Nelsen for ρ = −0.9,−0.4, 0.2 and 0.8 with n = 20
5 10 15 20
m value (minimun mean sample copula for m = 20)
0.0
0.1
0.2
0.3
0.4
0.5
Supremum Distance Example3.4(0.2,d=2)n20iters10000
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
5 10 15 20
m value (minimun sample mean copula for m = 20)
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7 Supremum Distance Example3.4(0.5,d=2)n20iters10000
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
5 10 15 20
m value (minimun mean sample copula for m = 20)
0.0
0.1
0.2
0.3
0.4
Supremum Distance Example3.4(0.7,d=2)n20iters2000
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
5 10 15 20
m value (minimun mean sample copula for m = 20)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
Supremum Distance Example3.4(0.95,d=2)n20iters2000
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
Figure 2.15: Example 3.4 of Nelsen for ρ = −0.0396, 0.5, 0.2591 and −0.7151 with n = 20
5 10 15 20
m value (minimun sample mean copula for m = 2)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
Supremum Distance Protocolo(d=2)n20iters10000
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
5 10 15 20
m value (minimun mean sample copula for m = 7)
0.00
0.05
0.10
0.15
0.20
0.25
Supremum Distance GumbelID(5,d=2)n20iters2000
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
5 10 15 20
m value (minimun mean sample copula for m = 7)
0.00
0.05
0.10
0.15
0.20
Supremum Distance GumbelID(10,d=2)n20iters2000
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
5 10 15 20
m value (minimun mean sample copula for m = 11)
0.00
0.05
0.10
0.15
0.20
Supremum Distance GumbelID(20,d=2)n20iters2000
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
Figure 2.16: Cop, Protocol ρ = .75 and GumbelID for ρ = −.9429,−.9854 and −.9963 with n = 20
34
5 10 15 20
m value (minimun mean sample copula for m = 4)
0.00
0.05
0.10
0.15
0.20
0.25
Supremum Distance GumbelDI(3,d=2)n20iters2000
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
5 10 15 20
m value (minimun mean sample copula for m = 7)
0.00
0.05
0.10
0.15
0.20
0.25
Supremum Distance GumbelDI(5,d=2)n20iters2000
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
5 10 15 20
m value (minimun mean sample copula for m = 7)
0.00
0.05
0.10
0.15
0.20
0.25
Supremum Distance GumbelDI(10,d=2)n20iters2000
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
5 10 15 20
m value (minimun mean sample copula for m = 11)
0.00
0.05
0.10
0.15
0.20
Supremum Distance GumbelDI(20,d=2)n20iters2000
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
Figure 2.17: Copulas GumbelDI for ρ = −.8487,−.9431,−.9855 and −.9963 with n = 20
5 10 15 20
m value (minimun mean sample copula for m = 4)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
Supremum Distance GumbelDD(3,d=2)n20iters2000
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
5 10 15 20
m value (minimun mean sample copula for m = 5)
0.00
0.05
0.10
0.15
0.20
Supremum Distance GumbelDD(5,d=2)n20iters2000
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
5 10 15 20
m value (minimun mean sample copula for m = 7)
0.00
0.05
0.10
0.15
0.20
Supremum Distance GumbelDD(10,d=2)n20iters2000
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
5 10 15 20
m value (minimun mean sample copula for m = 10)
0.00
0.05
0.10
0.15
0.20
Supremum Distance GumbelDD(20,d=2)n20iters2000
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
Figure 2.18: Copulas GumbelDD for ρ = .8494, .9434, .9854 and .9963 with n = 20
5 10 15 20
m value (minimun mean sample copula for m = 10)
0.00
0.05
0.10
0.15
0.20
Supremum Distance PlackettID(0.001,d=2)n20iters2000
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
5 10 15 20
m value (minimun mean sample copula for m = 2)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
Supremum Distance PlackettID(3,d=2)n20iters2000
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
5 10 15 20
m value (minimun mean sample copula for m = 3)
0.00
0.05
0.10
0.15
0.20
0.25
Supremum Distance PlackettID(10,d=2)n20iters2000
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
5 10 15 20
m value (minimun mean sample copula for m = 6)
0.00
0.05
0.10
0.15
0.20
0.25
Supremum Distance PlackettID(50,d=2)n20iters2000
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
Figure 2.19: Copulas PlackettID for ρ = 0.9880,−.3521,−.6537 and −.8780 with n = 20
5 10 15 20
m value (minimun mean sample copula for m = 10)
0.00
0.05
0.10
0.15
0.20
Supremum Distance PlackettDI(0.001,d=2)n20iters2000
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
5 10 15 20
m value (minimun mean sample copula for m = 2)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
Supremum Distance PlackettDI(3,d=2)n20iters2000
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
5 10 15 20
m value (minimun mean sample copula for m = 3)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
Supremum Distance PlackettDI(10,d=2)n20iters2000
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
5 10 15 20
m value (minimun mean sample copula for m = 6)
0.00
0.05
0.10
0.15
0.20
0.25
Supremum Distance PlackettDI(50,d=2)n20iters2000
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
Figure 2.20: Copulas PlackettDI for ρ = 0.9880,−.3521,−.6537 and −.8780 with n = 20
35
5 10 15 20
m value (minimun mean sample copula for m = 8)
0.00
0.05
0.10
0.15
0.20
Supremum Distance PlackettDD(0.001,d=2)n20iters2000
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
5 10 15 20
m value (minimun mean sample copula for m = 2)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
Supremum Distance PlackettDD(3,d=2)n20iters2000
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
5 10 15 20
m value (minimun mean sample copula for m = 3)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
Supremum Distance PlackettDD(10,d=2)n20iters2000
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
5 10 15 20
m value (minimun mean sample copula for m = 4)
0.00
0.05
0.10
0.15
0.20
0.25
Supremum Distance PlackettDD(50,d=2)n20iters2000
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
Figure 2.21: Copulas PlackettDD for ρ = −0.9881, .3517, .6532 and .8778 with n = 20
5 10 15 20
m value (minimun sample mean copula for m = 3)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
Supremum Distance Mix23(1.5,10,8,0.3,0.4,0.3,d=2)n20iters10000
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
5 10 15 20
m value (minimun mean sample copula for m = 2)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
Supremum Distance Mix23(15,15,6,0.4,0.5,0.1,d=2)n20iters10000
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
5 10 15 20
m value (minimun mean sample copula for m = 4)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
Supremum Distance Mix23(20,24,20,0.8,0.1,0.1,d=2)n20iters10000
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
5 10 15 20
m value (minimun sample mean copula for m = 2)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
Supremum Distance Mix23(1.5,10,10,0.8,0.1,0.1,d=2)n20iters10000
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
Figure 2.22: Mixt. of G, GID and GDI with ρ= -.54459, -.19608, .59829 and .18460 with n = 20
5 10 15 20
m value (minimun sample mean copula for m = 4)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
Supremum Distance Mix24(20,24,50,.3,.3,.4,d=2)n20iters10000
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
5 10 15 20
m value (minimun sample mean copula for m = 7)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
Supremum Distance Mix24(20,24,50,.1,.1,.8,d=2)n20iters10000
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
5 10 15 20
m value (minimun sample mean copula for m = 5)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
Supremum Distance Mix24(20,24,50,.1,.8,.1,d=2)n20iters10000
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
5 10 15 20
m value (minimun sample mean copula for m = 4)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
Supremum Distance Mix24(20,24,50,.8,.1,.1,d=2)n20iters10000
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
Figure 2.23: Mixt. of GID, Fr. and Joe with ρ = .39085, .79548, .77585 and -.600093 with n = 20
5 10 15 20
m value (minimun sample mean copula for m = 4)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
Supremum Distance Mix25(0.25,d=2)n20iters10000
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
5 10 15 20
m value (minimun sample mean copula for m = 2)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
Supremum Distance Mix25(0.5,d=2)n20iters10000
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
5 10 15 20
m value (minimun sample mean copula for m = 4)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
Supremum Distance Mix25(0.75,d=2)n20iters10000
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
5 10 15 20
m value (minimun mean sample copula for m = 9)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
Supremum Distance Mix25(0.9,d=2)n20iters2000
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
Figure 2.24: Mixt. of M2 and W2 with ρ = −0.5, 0, 0.5 and 0.8 with n = 20
36
Now we make a quick study of the variances for the twenty five families with n = 20 related to
all the Figures considerd above. In Figure 2.25, Figure 2.26, . . . , Figure 2.30 and Figure 2.31
we include the behavior of the variances of the statistics that measure the differences between
the supremum distance between the sample copula of order m and the real copula, and also the
supremum distance between the empirical copula and the real copula. We chose only one value
of the parameter in everyone of the previous Figures. For each family, the chosen parameter best
describes the behavior of the variance, leaving out the limit cases. In all graphs we give the value
of ρ.
First we observe that from Figure 2.25 the variances for the Ali-Mikhail-Haq family behave like the
variance given for the product copula Π2 in Figure 2.27, this fact follows from what we explained
before for Figures 2.1 and 2.2. Observe that the biggest variance is given for m = 2, but the
difference of the variance of the sample copula and the variance for the empirical copula in this
case is only of magnitude 0.0005. For larger values of m, the sample copula has practically the
same variance as the empirical copula.
In the cases of the Clayton, Frank, Gumbel, Joe, Normal, Plackett and t-Student families, the
variance of the sample copula of order m for 10 ≤ m ≤ 20 = n is close to the variance of the
empirical copula, and in some cases the variances of the sample copula of order m are smaller than
the ones reported for the empirical copula for some values of m, see Figure 2.25, Figure 2.26 and
Figure 2.27.
In the case of singular copulas such as the circular distribution, Example 3.3, Example 3.4 and
Example 3.5 we see that in average the variance of the sample copula of order m is smaller than
the variance for the empirical case.
For the protocol copula, see Figure 2.29, we observe that the oscillating behavior of the variances
for 2 ≤ m ≤ n = 20 follows the pattern of the mean, minimum and maximum given in Figure 2.16,
giving greater variances to even values of m and smaller variances for odd values of m.
In the case of considering increasing and decreasing transformations in the Gumbel and Plackett
families, we observe that the variances graphs in Figure 2.29 and Figure 2.30 have a similar be-
havior when we keep the parameter fixed in the cases GumbelID, GumbelDI and GumbelDD, and
also fixed in the cases PlackettID, PlackettDI and PlackettDD. If we observe that the cases ID, DD
and DI correspond of rotations of 90, 180 and 270 degrees of the original copula around the point
(1/2, 1/2), then we have that there is a certain invariance of the graphs, this invariance can also be
observed in Figure 2.16, Figure 2.17 and Figure 2.18, and in Figure 2.19, Figure 2.20 and Figure
2.21.
37
For the case of mixtures we denote Mix23 the mixture of Gumbel, GumbelID and GumbelDI,
Mix24 is the mixture of GID, Frank and Joe, in this two cases the first three numbers correspond to
the parameters of each member of the mixture, and the last three correspond to the weights. Mix25
is the mixture of M2 and W2. In these cases we observe that the variances for the sample copula of
order m remain below the variances of the empirical copula almost in all cases.
38
5 10 15 20
m value
0.000
0.001
0.002
0.003
0.004
Variances AMH(-0.5,d=2)n20iters10000
sample copula empirical copula
5 10 15 20
m value
0.000
0.001
0.002
0.003
0.004
Variances AMH(0.5,d=2)n20iters10000
sample copula empirical copula
5 10 15 20
m value
0.000
0.001
0.002
0.003
0.004
Variances Clayton(-0.5,d=2)n20iters2000
sample copula empirical copula
5 10 15 20
m value
0.000
0.001
0.002
0.003
0.004
Variances Clayton(5,d=2)n20iters2000
sample copula empirical copula
Figure 2.25: Var. of AMH(-0.148), AMH(0.192), Clay.(-0.467) and Clay.(0.844) with n = 20
5 10 15 20
m value
0.000
0.001
0.002
0.003
0.004
Variances Frank(-10,d=2)n20iters2000
sample copula empirical copula
5 10 15 20
m value
0.000
0.001
0.002
0.003
0.004
Variances Frank(7,d=2)n20iters2000
sample copula empirical copula
5 10 15 20
m value
0.000
0.001
0.002
0.003
0.004
Variances Gumbel(15,d=2)n20iters2000
sample copula empirical copula
5 10 15 20
m value
0.000
0.001
0.002
0.003
0.004
Variances Joe(5,d=2)n20iters2000
sample copula empirical copula
Figure 2.26: Var. of Fr.(-0.860), Fr.(0.763), Gum.(0.993) and Joe(0.854) with n = 20
5 10 15 20
m value
0.000
0.001
0.002
0.003
0.004
Variances Normal(-0.5,d=2)n20iters10000
sample copula empirical copula
5 10 15 20
m value
0.000
0.001
0.002
0.003
0.004
Variances Plackett(10,d=2)n20iters2000
sample copula empirical copula
5 10 15 20
m value
0.000
0.001
0.002
0.003
0.004
Variances Producto(d=2)n20iters10000
sample copula empirical copula
5 10 15 20
m value
0.000
0.001
0.002
0.003
0.004
Variances Tstudent(-0.6,d=2)n20iters2000
sample copula empirical copula
Figure 2.27: Var. of Norm.(-0.482), Plack.(0.654), Prod.(0) and t-Student(-0.582) with n = 20
5 10 15 20
m value
0.000
0.001
0.002
0.003
0.004
Variances Circular(d=2)n20iters10000
sample copula empirical copula
5 10 15 20
m value
0.000
0.001
0.002
0.003
0.004
Variances Example3.5(d=2)n20iters10000
sample copula empirical copula
5 10 15 20
m value
0.000
0.001
0.002
0.003
0.004
Variances Example3.3(0.6,d=2)n20iters10000
sample copula empirical copula
5 10 15 20
m value
0.000
0.001
0.002
0.003
0.004
Variances Example3.4(0.5,d=2)n20iters10000
sample copula empirical copula
Figure 2.28: Var. of Circ.(0), Ex.3.5(.286), Ex.3.3(0.2) and Ex.3.4(0.5) with n = 20
39
5 10 15 20
m value
0.000
0.001
0.002
0.003
0.004
Variances Protocolo(d=2)n20iters10000
sample copula empirical copula
5 10 15 20
m value
0.000
0.001
0.002
0.003
0.004
Variances GumbelID(10,d=2)n20iters2000
sample copula empirical copula
5 10 15 20
m value
0.000
0.001
0.002
0.003
0.004
Variances GumbelDI(10,d=2)n20iters2000
sample copula empirical copula
5 10 15 20
m value
0.000
0.001
0.002
0.003
0.004
Variances GumbelDD(10,d=2)n20iters2000
sample copula empirical copula
Figure 2.29: Var. of Protocol(.75), GID(-.985), GDI(-.985) and GDD(.985) with n = 20
5 10 15 20
m value
0.000
0.001
0.002
0.003
0.004
Variances PlackettID(10,d=2)n20iters2000
sample copula empirical copula
5 10 15 20
m value
0.000
0.001
0.002
0.003
0.004
Variances PlackettDI(10,d=2)n20iters2000
sample copula empirical copula
5 10 15 20
m value
0.000
0.001
0.002
0.003
0.004
Variances PlackettDD(10,d=2)n20iters2000
sample copula empirical copula
Figure 2.30: Var. of PID(-.653), PDI(-.653) and PDD(.653) with n = 20
5 10 15 20
m value
0.000
0.001
0.002
0.003
0.004
Variances Mix23(20,24,20,0.8,0.1,0.1,d=2)n20iters10000
sample copula empirical copula
5 10 15 20
m value
0.000
0.001
0.002
0.003
0.004
Variances Mix24(20,24,50,.1,.8,.1,d=2)n20iters10000
sample copula empirical copula
5 10 15 20
m value
0.000
0.001
0.002
0.003
0.004
Variances Mix25(0.5,d=2)n20iters10000
sample copula empirical copula
Figure 2.31: Var. of Mix23(0.5977), Mix24(0.7758) and Mix25(0)
We also performed several simulations for sample sizes n = 30 and n = 50 in these two cases we
obtained very similar graphs to the ones obtained above in Figures one to thirty one, but of course
in different scales. We do not present all these results because the differences are pretty much
negligible.
In the next Subsection we will see that we can extend the previous results for higher dimensions.
2.4 Simulation study: dimension three
In this subsection, we study eleven different families in dimension d = 3, which include families
of copulas such as: Clayton, Frank, Gumbel, Normal, Product and t-Student. We also included
the copula M3 and its transformations by increasing and decresing families to obtain 3-copulas
with support in every diagonal of I3, these are examples of singular copulas. We also give a
40
3-dimensional version of the protocol copula to see what happens when we have an absolutely
continuous 3-copula with support [0, 1/2]3 ∪ (1/2, 1]3. We also include just as an interesting re-
ference a 3-copula denoted by C =“W3” such that C(2/3, 2/3, 2/3) = 0, that is, it behaves like W3,
which is not a 3-copula at the point (2/3, 2/3, 2/3). Finally we include two families of mixtures of
3-copulas to study the behavior under asymmetric 3-copulas.
For the product copula Π3 in d = 3, see Figure 2.34 first graph, the behavior of the difference
between the supremum distances is quite similar to the case d = 2 with n = 30, see Figure 11.
Hence, we have that C30
m is a better approximation than C30 for every 2 ≤ m ≤ n = 30. Observe
that the variances have the same behavior as in dimension 2, see Figure 2.27 and Figure 2.40.
As can be seen in Figure 2.32, Figure 2.33, Figure 2.34, Figure 2.35 and Figure 2.36, the cases
Clayton, Frank, Gumbel, Normal and Clayton have similar behavior, and at least at these graphs,
C30
m is a better approximation than C30 for every 6 ≤ m ≤ n = 30. The same similarity holds for the
graphs of the variances in Figure 2.40, Figure 2.41 and Figure 2.42.
In Figure 2.37 we show the results for singular copulas, the first graph corresponds to the copula
M3, the second, third and fourth graphs for M3IID,M3IDI and M3DII, that is, if the vector (U,V,W)
has copula M3. Then the vector (U,V, 1−W) has copula M3IID, the vector (U, 1−V,W) has copula
M3IDI and the vector (1 − U,V,W) has copula M3DII, these three copulas have supports on the
other three diagonals of the unit cube [0, 1]3, and its existence is guaranteed by an easy extension
of Theorem 2.4.4 in [37]. As in the case of dimension d = 2, the variances in the four cases are
zero.
The case that we called protocol consists of a 3-copula which has incomplete support given by
[0, 1/2]3 ∪ (1/2, 1]3, with uniform masses on each cube. We can see from the first graph in Figure
2.38, that C30
2
is a better approximation than C30 and C30
4
is a better approximation than C30, but
C30
3
is not a better approximation than C30 and C30
5
is not a better approximation than C30, that is
the oscillating pattern is also present in dimension d = 3. In this case C30
m is a better approximation
than C30 for every 6 ≤ m ≤ 30 = n. In the case of the variance, see Figure 2.41, this oscillation is
also evident with larger variances for even values of m and smaller variances for odd values of m.
The case Mixture 9 is just a convex combination of a Clayton, a Gumbel and a Frank, in Figure
2.38 we present the results when the corresponding parameters are 10, 15 and 20 respectively, and
with weights 0.2, 0.3 and 0.5 in different orders. As we can see C30
m is a better approximation than
C30 for every 5 ≤ m ≤ 30 = n. The variances are almost always below the variance of the empirical
copula, see Figure 2.42.
41
The case Mixture 10 which corresponds to a convex combination of Gumbel, GumbeIID, Gumbe-
lIDI and GumbelDII, defined as in the case of M3 above. This family is highly assymetric in all
directions. We present two cases the first when the parameters are 5, 10, 15 and 20 with weights
0.1, 0.2, 0.3 and 0.4 in three different orders, and the second when all the parameters are equal to
10 and all the weights are equal to 0.25. The results are presented in Figure 2.39, and in these cases
we have that Cm
30
is a better approximation than C30 for every 2 ≤ m ≤ 30 = n. The variances are
again almost always below the variance of the empirical copula, see Figures 2.40, 2.41 and 2.42.
The variance of the copula denoted by “W3” is presented in Figure 2.40 last graph.
42
5 10 15 20 25 30
m value (minimun sample mean copula for m = 2)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
Supremum Distance Clayton(0.5,d=3)n30iters1500
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
5 10 15 20 25 30
m value (minimun sample mean copula for m = 3)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
Supremum Distance Clayton(1,d=3)n30iters1500
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
5 10 15 20 25 30
m value (minimun sample mean copula for m = 5)
0.00
0.05
0.10
0.15
0.20
0.25
Supremum Distance Clayton(5,d=3)n30iters1500
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
5 10 15 20 25 30
m value (minimun sample mean copula for m = 10)
0.00
0.05
0.10
0.15
0.20
0.25
Supremum Distance Clayton(20,d=3)n30iters1500
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
Figure 2.32: Clayton d = 3 for ρ = 0.2955, 0.4784, 0.8848 and 0.9869 with n = 30
5 10 15 20 25 30
m value (minimun sample mean copula for m = 2)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
Supremum Distance Frank(1,d=3)n30iters2000
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
5 10 15 20 25 30
m value (minimun sample mean copula for m = 3)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
Supremum Distance Frank(5,d=3)n30iters2000
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
5 10 15 20 25 30
m value (minimun sample mean copula for m = 4)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
Supremum Distance Frank(8,d=3)n30iters2000
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
5 10 15 20 25 30
m value (minimun sample mean copula for m = 5)
0.00
0.05
0.10
0.15
0.20
0.25
Supremum Distance Frank(10,d=3)n30iters2000
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
Figure 2.33: Frank d = 3 for ρ = 0.1644, 0.6434, 0.8035 and 0.8602 with n = 30
5 10 15 20 25 30
m value (minimun sample mean copula for m = 2)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
Supremum Distance Producto(d=3)n30iters10000
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
5 10 15 20 25 30
m value (minimun sample mean copula for m = 4)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
Supremum Distance Gumbel(2,d=3)n30iters2000
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
5 10 15 20 25 30
m value (minimun sample mean copula for m = 6)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
Supremum Distance Gumbel(5,d=3)n30iters2000
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
5 10 15 20 25 30
m value (minimun sample mean copula for m = 10)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
Supremum Distance Gumbel(10,d=3)n30iters2000
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
Figure 2.34: Product (0)and Gumbel d = 3 for ρ = 0.6828, 0.9430 and 0.9855 with n = 30
5 10 15 20 25 30
m value (minimun sample mean copula for m = 3)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
Supremum Distance Normal(.2,.5,.7,d=3)n30iters2000
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
5 10 15 20 25 30
m value (minimun sample mean copula for m = 3)
0.00
0.05
0.10
0.15
0.20
0.25
Supremum Distance Normal(-.6,-.85,.1,d=3)n30iters2000
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
5 10 15 20 25 30
m value (minimun sample mean copula for m = 3)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
Supremum Distance Normal(-.8,-.8,.29,d=3)n30iters2000
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
5 10 15 20 25 30
m value (minimun sample mean copula for m = 10)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
Supremum Distance Normal(.99,.99.99,d=3)n30iters2000
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
Figure 2.35: Normal d = 3 for θ = (.2, .5, .7), (−.6,−.85, .1), (−.8,−.8, .29) and (.99, .99, .99)
43
5 10 15 20 25 30
m value (minimun sample mean copula for m = 2)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
Supremum Distance Tstudent(0,0,d=3)n30iters2000
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
5 10 15 20 25 30
m value (minimun sample mean copula for m = 3)
0.00
0.05
0.10
0.15
0.20
0.25
Supremum Distance Tstudent(.5,-.5,d=3)n30iters2000
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
5 10 15 20 25 30
m value (minimun sample mean copula for m = 2)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
Supremum Distance Tstudent(.4,-.6799,d=3)n30iters2000
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
5 10 15 20 25 30
m value (minimun sample mean copula for m = 4)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
Supremum Distance Tstudent(.7071,0,d=3)n30iters2000
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
Figure 2.36: t-Student d = 3 for θ = (0, .0), (.5,−.5), (.4,−.6799) and (.7071, 0) with n = 30
5 10 15 20 25 30
m value (minimun sample mean copula for m = 30)
0.00
0.05
0.10
0.15
0.20
0.25
Supremum Distance M(d=3)n30iters2
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
5 10 15 20 25 30
m value (minimun sample mean copula for m = 30)
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
Supremum Distance MIID(2,d=3)n30iters2
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
5 10 15 20 25 30
m value (minimun sample mean copula for m = 30)
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
Supremum Distance MIDI(3,d=3)n30iters2
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
5 10 15 20 25 30
m value (minimun sample mean copula for m = 30)
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
Supremum Distance MDII(4,d=3)n30iters2
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
Figure 2.37: M3, M3IID, M3IDI and M3DII with n = 30
5 10 15 20 25 30
m value (minimun sample mean copula for m = 2)
0.0
0.1
0.2
0.3
0.4
0.5
Supremum Distance Protocolo(d=3)n30iters2000
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
5 10 15 20 25 30
m value (minimun sample mean copula for m = 7)
0.00
0.05
0.10
0.15
0.20
0.25
Supremum Distance Mix9(10,15,20,.3,.2,.5,d=3)n30iters2000
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
5 10 15 20 25 30
m value (minimun sample mean copula for m = 9)
0.00
0.05
0.10
0.15
0.20
0.25
Supremum Distance Mix9(10,15,20,.3,.5,.2,d=3)n30iters2000
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
5 10 15 20 25 30
m value (minimun sample mean copula for m = 8)
0.00
0.05
0.10
0.15
0.20
0.25
Supremum Distance Mix9(10,15,20,.5,.3,.2,d=3)n30iters2000
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
Figure 2.38: Protocol and Mixture 9 d = 3 for θ = (10, 15, 20) for different weights with n = 30
5 10 15 20 25 30
m value (minimun sample mean copula for m = 4)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
Supremum Distance Mix10(5,10,15,20,.1,.2,.3,.4,d=3)n30it2000
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
5 10 15 20 25 30
m value (minimun sample mean copula for m = 4)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
Supremum Distance Mix10(5,10,15,20,.2,.4,.3,.1,d=3)n30it2000
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
5 10 15 20 25 30
m value (minimun sample mean copula for m = 4)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
Supremum Distance Mix10(5,10,15,20,.4,.3,.2,.1,d=3)n30it2000
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
5 10 15 20 25 30
m value (minimun sample mean copula for m = 4)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
Supremum Distance Mix10(10,10,10,10,.25,.25,.25,.25,d=3)n30it2000
sample copula
empirical copula
sample minimun
sample maximum
empirical minimun
empirical maximum
Figure 2.39: Mixture 10 d = 3 for θ = (5, 10, 15, 20) and (10, 10, 10, 10) for different weights
44
5 10 15 20 25 30
m value
0.000
0.001
0.002
0.003
0.004
Variances Clayton(5,d=3)n30iters1500
sample copula empirical copula
5 10 15 20 25 30
m value
0.000
0.001
0.002
0.003
0.004
Variances Frank(8,d=3)n30iters2000
sample copula empirical copula
5 10 15 20 25 30
m value
0.000
0.001
0.002
0.003
0.004
Variances Producto(d=3)n30iters10000
sample copula empirical copula
5 10 15 20 25 30
m value
0.000
0.001
0.002
0.003
0.004
Variances W(1,d=3)n30iters2000
sample copula empirical copula
Figure 2.40: Variances of Clayton(ρ = 0.8848), Frank(ρ = 0.8035), Product and “W3” when
n = 30
5 10 15 20 25 30
m value
0.000
0.001
0.002
0.003
0.004
Variances Gumbel(10,d=3)n30iters2000
sample copula empirical copula
5 10 15 20 25 30
m value
0.000
0.001
0.002
0.003
0.004
Variances Normal(-.8,-.8,.29,d=3)n30iters2000
sample copula empirical copula
5 10 15 20 25 30
m value
0.000
0.001
0.002
0.003
0.004
Variances Protocolo(d=3)n30iters2000
sample copula empirical copula
Figure 2.41: Var. of Gumb.(ρ = 0.9855), Normal(−0.8,−0.8, 0.29) and Protocol copula when
n = 30
5 10 15 20 25 30
m value
0.000
0.001
0.002
0.003
0.004
Variances Tstudent(.7071,0,d=3)n30iters2000
sample copula empirical copula
5 10 15 20 25 30
m value
0.000
0.001
0.002
0.003
0.004
Variances Mix9(10,15,20,.5,.3,.2,d=3)n30iters2000
sample copula empirical copula
5 10 15 20 25 30
m value
0.000
0.001
0.002
0.003
0.004
Variances Mix10(5,10,15,20,.4,.3,.2,.1,d=3)n30iters2000
sample copula empirical copula
Figure 2.42: Variances of t-St.(5), Mix9(10, 15, 20, .3, .2, .5) and Mix10(5, 10, 15, 20, .4, .3, .2, .1)
As we have seen in all these simulations not only in dimension d = 3, but also in dimension d = 2,
we can say that there exists at least one m such Cn
m is a better approximation than Cn, and that this
value of m is smaller than n.
The results that we obtained for n = 20, which are not presented here, have many features in
common with the case n = 30.
It is obvious that these results can be extended to higher dimensions. In fact, the programs that we
wrote in language R can be easily extended to at least dimension d = 5 without any problems.
45
2.5 A method for estimating an adequate m
From all the examples in Section 3 we can observe that even for a small value of n, that is, n = 20
in dimension d = 2, we can find an m with 2 ≤ m ≤ n, such that Cn
m the sample copula of order
m is a better approximation than the empirical copula Cn. In fact, in all cases, we can also find
a value of m that minimizes the expected value of the difference between Cn
m and the real copula
C, observe all the comments between parenthesis below each of the graphs in Figures 2.1 through
Figure 2.24. It is quite important to observe that this minimum is reached at m = 2 when the real
copula C is “close” to the product copula Π2, and that this minimum increases as the real copula C
approaches the Frechet-Hoeffding bounds M2 and W2, in fact, as observed in Figure 2.13, in these
two cases the minimum is reached at m = n, this also holds for the Example 3.4 of Nelsen’s book
which is a singular copula, see Figure 2.15.
As observed in Section 3, in the case of dimension d = 2, when we have an absolutely continuous
copula with complete support, which by the way is the most interesting case in applications, we
have a function that relates the Spearman’s ρ with the value of m that minimizes the expected value
of the difference between Cn
m and the real copula C.
In Figure 2.43 we graph the value of m which minimizes the expected value of the supremum
distance between the sample copula Cn
m and the real copula C, for sample sizes n = 20, 30 and
n = 50, for the families of absolutely continuous copulas with complete support which include
AMH, Clayton, Gumbel, Frank, Joe, Plackett, Normal, t, Plackett ID, Plackett DI, Plackett DD
and Gumbel DD, for different values of ρ between ρ = 0.1 and ρ = .99. As can be seen from these
graphs, for ρ fixed, the value of m is quite stable and it is always smaller than the sample size n.
This result provides a natural way to give an estimator of the parameter m based on the sample
Spearman’s ρ.
Now we propose a method to estimate m for the sample copula of order m based on a random
sample of size n:
• Obtain the modified sample.
• Graph the points of the modified sample. If the data does not follow a clear pattern which
indicates that the copula C is a discrete copula then
• Estimate the Spearman’s rho using the sample value ρ̃ of the Spearman’s ρ.
• If |ρ̃| ≤ 0.25 we can take m = 2 for any sample size.
• If |ρ̃| ≥ .95 we have to take a large value of m. Let us say 10 ≤ m ≤ 15.
46
• If 0.25 < |ρ̃| < 0.95 take 3 ≤ m ≤ 10, with a linear approximation which depends on the
value of ρ̃.
Cl AMH Gu Fr J P T N PID PDI PDD GDD
family
0
5
10
15
20
m
-m
in
ρ=0.1
n=20 n=30 n=50
Cl AMH Gu Fr J P T N PID PDI PDD GDD
family
0
5
10
15
20
m
-m
in
ρ=0.25
n=20 n=30 n=50
Cl Gu Fr J P T N PID PDI PDD GDD
family
0
5
10
15
20
m
-m
in
ρ=0.5
n=20 n=30 n=50
Cl Gu Fr J P T N PID PDI PDD GDD
family
0
5
10
15
20
m
-m
in
ρ=0.75
n=20 n=30 n=50
Cl Gu Fr J P T N PID PDI PDD GDD
family
0
5
10
15
20
m
-m
in
ρ=0.9
n=20 n=30 n=50
Cl Gu Fr J P T N PID PDI PDD GDD
family
0
5
10
15
20
m
-m
in
ρ=0.95
n=20 n=30 n=50
Cl Gu Fr J P T N PID PDI PDD GDD
family
0
5
10
15
20
m
-m
in
ρ=0.99
n=20 n=30 n=50
Figure 2.43
If the modified sample shows a pattern which indicates a singular support we may need to take
larger values of m, but not larger than 15.
We also observe that for d ≥ 3 the same methodology can be used for Archimedean copulas, since
all of their margins have the same parameter, hence, the same value of Spearman’s ρ for each pair
of random variables. This behavior was observed in Figure 2.32, Figure 2.33 and Figure 2.34.
47
For dimension d ≥ 3 we are still working on the possibility of estimating m using the averages of
all the estimated Spearman’s ρ for every pair of random variables.
Also, for dimension d ≥ 3, we know from all the Figures, that there exists a value 2 ≤ m0 ≤ n
such that for every m0 ≤ m ≤ n, Cn
m is a better approximation than Cn, of course, the minimum
value of m0 is among them. We would like to find a method of estimation of m, given a fix sample
(X
1
, . . . , X
n
) of size n from an unknown continuous joint distribution function H with copula C,
such that m0 ≤ m ≤ n. Since the copula C is also unknown, we will use all the remarks that we have
made in order to establish a “measure of discrepancy” between the sample and the product copula
Π2, which may be used to estimate the value of m. Besides, from Figure 2.32 through Figure 2.42
we observe that all the previous comments apply to n = 30 in dimension d = 3, this also holds
when n = 20.
We start by defining our proposal of the “measure of discrepancy” in any dimension d ≥ 2 for n a
fixed positive integer.
Definition 2.18 Let 2 ≤ m ≤ n and let (X
1
, . . . , X
n
) be a random sample from a random vector
X of dimension d ≥ 2 with continuous joint distribution H or d-copula C. Let Un = (Y
1
, . . . ,Y
n
)
be the corresponding modified sample. Define (Qm
i
)i∈Id
m
, (Rm
i
)i∈Id
m
, s
n,(m)
i1,...,id
, S n
m and Cn
m as in equations
(15), (16), (17), (18) and (19) respectively. Let Πd the product d-copula, since Cn
m the sample
d-copula of order m is always a d-copula, define for every (i1, . . . , id) ∈ Id
m
VCn
m
(Qm
i1,...,id
) = s
n,(m)
i1,...,id
and VΠd (Qm
i1,...,id
) = λd(Qm
i1,...,id
), (30)
where λd is the Lebesgue measure on ( RI d,B( RI d). We define
dTV(Cn
m,Π
d) =
1
2
∫
[0,1]2
| fCn
m
− fΠd |dλd =
∑
i∈(Im)d
∣
∣
∣
∣
VCn
m
(Qm
i
) − VΠd (Qm
i
)
∣
∣
∣
∣
2
, (31)
We first observe that from equations (30), (17) and (31) we have that
dTV(Cn
m,Π
d) =
∑
i∈(Im)d
∣
∣
∣
∣
s
n,(m)
i
− λ(Qm
i
)
∣
∣
∣
∣
2
=
∑
i∈(Im)d
∣
∣
∣
∣
∣
card(Rm
i
∩Un)
n
− λ(Qm
i
)
∣
∣
∣
∣
∣
2
. (32)
So, if we assume that n = md and that card(Rm
i
∩ Un) = 1 for every i ∈ Id
m, we have from equation
(32) that dTV(Cn
m,Π
d) = 0.
48
Now, we assume that the sample of size n is taken from the the copula Md, then clearly the modified
sample is given by Un = (Y
1
, . . . ,Y
n
), where Y j = ( j/n, j/n, . . . , j/n) for every j ∈ {1, 2, . . . , n}.
First, if m divides n then we know from Theorem 2.14, part iii) that (Qm
i
)i∈Id
m
coincides with (Rm
i
)i∈Id
m
,
so, λd(Qm
i
) = (1/m)d for every i ∈ Id
m. Since the modified sample lies on the main diagonal of the
unit d cube Id, then we have from equation (21) that for every i ∈ Im if Rm
i=(i,i,...,i)
= [(i − 1)/m, i/m]d
then
card(Ri∩Un)
n
= (n/m)/n = 1/m. Hence, using equation (32) we have that
dTV(Cn
m,Π
d) =







m
(
1
m
(
md−1 − 1
md−1
))
+ (md − m)
(
1
m
)d







/
2 = 2
(
md−1 − 1
md−1
)/
2 = 1 − 1
md−1
.
Second, if m does not divide m then Theorem 2.14 part i) we have that
Qm
i=(i,i,...,i) = [(1/n) ⌊((i − 1) · n)/m⌋ , (1/n) ⌊(i · n)/m⌋]d
for every i ∈ Im. So, in this case, λd(Qm
i=(i,i,...,i)
) = (1/(nd)) (⌊(i · n)/m⌋ − ⌊((i − 1) · n)/m⌋)d and
VCn
m
(Qm
i=(i,i,...,i)
) = (1/n) (⌊i/m⌋ − ⌊(i − 1)/m⌋) for every i ∈ Im. So, using the fact that the sum of all
the volumes of the d-boxes in the partition (Qm
i
)i∈Id
m
is one, it is easy to see that
dTV(Cn
m,Π
d) =
(
md−1
md−1 − 1
)







1 −
m
∑
i=1
1
nd
(⌊(i · n)/m⌋ − ⌊((i − 1) · n)/m⌋)d







≤
(
md−1
md−1 − 1
) (
1 − m
(
1
md
))
= 1,
where the inequality follows from Lagrange multiplier applied to the function h(x1, . . . , xm) =
1 − (xd
1
+ · · · + xd
m) with conditions x1 + x2 + . . . + xm = 1 and x1, . . . , xm ≥ 0.
Hence, if we change the denominator 2 by Km,d = 2(1 − 1/md−1) in equation 31, we have that
0 ≤ dTV(Cn
m,Π
d) ≤ 1, and this what we will do in what follows.
In order to see how this density difference works, we show the averages of 10000 simulations when
the sample size is n = 100 and we evaluate ηm,2(C100
m ) for m between 2 and 10 for several of the
families studied in Section 3, we use only values of ρ ≥ 0, but the results for ρ < 0 follow the same
pattern.
In Figures 2.44 and 2.45, we have the results for the families Clayton,Frank, Gumbel, Plackett,
Normal and t-Student. It can be easily observed that for all these families of copulas which are
49
absolutely continuous and with complete support. the graphs are almost identical, which indicates
that they can be used to estimate the value of the order m of the sample d-copula. We have similar
results for d = 3 and we are working in a resampling method to estimate the graph given a fixed
sample of size n.
As can be see in Figure 2.46, the graphs for singular copulas can be quite different as the absolutely
continuous case, and it depends strongly on the support of the copula. But, as observed above, in
these cases we always need larger values of m. Observe that for the last graph which corresponds
to the copula M2, the graph is the constant 1, as expected.
50
2 4 6 8 10
m value
0.0
0.2
0.4
0.6
0.8
1.0
Positive Rho (Clayton n = 100, N = 10000)
2 4 6 8 10
m value
0.0
0.2
0.4
0.6
0.8
1.0
Positive Rho (Frank n = 100, N = 10000)
2 4 6 8 10
m value
0.0
0.2
0.4
0.6
0.8
1.0
Positive Rho (Gumbel n = 100, N = 10000)
Figure 2.44: Averages of ηm,2 for the Clayton, Frank and Gumbel families with positive ρ
2 4 6 8 10
m value
0.0
0.2
0.4
0.6
0.8
1.0
Positive Rho (Plackett n = 100, N = 10000)
2 4 6 8 10
m value
0.0
0.2
0.4
0.6
0.8
1.0
Positive Rho (Normal n = 100, N = 10000)
2 4 6 8 10
m value
0.0
0.2
0.4
0.6
0.8
1.0
Positive Rho (Tstudent n = 100, N = 10000)
Figure 2.45: Averages of ηm,2 for the Plackett, Normal and tStudent families with positive ρ
2 4 6 8 10
m value
0.0
0.2
0.4
0.6
0.8
1.0
Positive Rho (Example3.3 n = 100, N = 10000)
2 4 6 8 10
m value
0.0
0.2
0.4
0.6
0.8
1.0
Positive Rho (Example3.4 n = 100, N = 10000)
2 4 6 8 10
m value
0.0
0.2
0.4
0.6
0.8
1.0
Positive Rho (M n = 100, N = 10000)
Figure 2.46: Averages of ηm,2 for the Example 3.3, Example 3.4 and M2 families with positive ρ
51
A last very important result is the following: Recall that the total variation distance of two proba-
bility measures P and Q in ( RI d,B), where B denotes the Borel σ-algebra, with Radon-Nykodim’s
derivatives fP and fQ is defined by dTV(P,Q) = supA∈B |P(A) − Q(A)| = 1
2
∫
RI
d | fP − fQ|dλ ≤ 1,
where λ is the Lebesgue measure in ( RI d,B). Let us assume that we take a random sample from Πd
the independent copula of size n ≥ 2 and we we take m = n in the definition of the sample copula,
which corresponds to the linear B-spline copula construction in [43]. Since in this case, we are
considering the uniform partition of order n of [0, 1]d, that is, (Rn
i
)i∈Id
n
, using part v) of Theorem
2.14, we have that only n of the d-boxes in the uniform partition have density nd−1 and the remain-
ing boxes have density 0. Let J be the subset of n indices i ∈ Id
n with positive density. Then, using
the above definition we have that
dTV(Πd,Cn
n) =
1
2
∫
[0,1]d
| fΠd − fCn
n
|dλ
=
1
2










∑
i∈J
∫
Rn
i
|1 − nd−1|dλ +
∑
i∈(Id
n \J)
∫
Rn
i
|1 − 0|dλ










=
1
2
(
n(nd−1 − 1)
nd
+
(nd − n)
nd
)
=
1
2
(
1 − 1
nd−1
+ 1 − 1
nd−1
)
= 1 − 1
nd−1
. (33)
The last equality implies that, with probability one, if we let n go to infinity then dTV(Πd,Cn
n) ↑ 1.
Hence, it can be thought as an “anti” Glivenko-Cantelli’s result. Even more, if we let the dimension
d go to infinity for any fixed n ≥ 2 then again dTV(Πd,Cn
n) ↑ 1. This argument tell us that using
m = n is not a really good option at all.
On the other hand, if we take d = 2, m = 2 and we assume that the sample size n is a multiple of
md = 4, and that we are sampling from Π2. Then with positive probability ((n/2)!)4/(n! · ((n/4)!)4),
see [26], we have that each 2-box of the uniform partition of order m = 2 has exactly n/4 points.
So, using Theorem 2.14, we have that the density of the sample copula of order 2 is one on each
2-box of the uniform partition. But in this case we have that dTV(C2
n,Π
2) = 0. In Figure 11 we can
see that when n = 20 the minimum value of the simulations attains 0, when m = 2. In fact, it is not
difficult to see that 0 ≤ dTV(C2
n,Π
2) ≤ 1/2 with probability one. Of course, the above argument
works also when d > 2, m ≥ 2 and the sample size n is a multiple of md.
52
Finally, we have observed using simulations, that the total variation distance may be used to mea-
sure the distance between Cm
n , the d-sample copula of order m, and C(m), the checkerboard approx-
imation of size m, and that when the sample size increases to infinity as a multiple of m, then we
obtain a Glivenko-Cantelli’s theorem. Observe that this happens even using the strongest distance,
that is, the total variation, which in this case is surprisingly easy to evaluate.
In Tables 1 to 7, we present the results of evaluating the total variation distance between Cm
n the
sample copula of order m = 10 and C(10) the checkerboard approximation, with 1000 simulations
of sample sizes varying between n = 10 and n = 50000, the rows include basic statistics such
as mean, variance, minimum and maximum. We used some of the absolutely continuous families
in Section 3, specifically the AMH, Clayton, Frank, Gumbel and Plackett copulas with different
values of Spearman’s ρ. We also include with m = 15 Π2 the product copula in dimension 2, with
n varying from 15 to 60000, and also, Π3 the product copula in dimension 3 when m = 2, when n
varies from n = 2 up to n = 49152. As can be seen in all cases, the statistics mean, minimum and
maximum decrease to zero as n increases, and the variance also decrease to zero for n large, which
gives evidence of the existence of a Glivenko-Cantelli’s theorem for the total variation distance.
Observe also, that the first columns in Table 6 and Table 7 agree with the result given in equation
(33). Even more, in the two last tables it is obvious that the checkerboard approximations coincides
with the real copulas, that is, Π2 and Π3, so, we have a real Glivenko-Cantelli’s theorem between
Cm
n and C the true copula.
Table 1 Total Variation Distance for AMH (ρ = 0.3451) with m = 10
n 10 100 500 1000 2000 10000 50000
mean 0.8874 0.3532 0.1573 0.1108 0.0786 0.0350 0.0156
variance 0.00015 0.00079 0.00017 0.00008 0.00004 8 e-06 1 e-06
minimum 0.8577 0.2605 0.1189 0.0758 0.0582 0.0248 0.0118
maximum 0.9244 0.4879 0.2037 0.1443 0.1032 0.0436 0.0204
Table 2 Total Variation Distance for Clayton (ρ = 0.9582) with m = 10
n 10 100 500 1000 2000 10000 50000
mean 0.5915 0.1757 0.0788 0.0555 0.0389 0.0175 0.0078
variance 0.00720 0.00090 0.00014 0.00007 0.00003 7 e-06 1 e-06
minimum 0.4521 0.1013 0.0459 0.0304 0.02236 0.0104 0.0043
maximum 0.8579 0.2835 0.1153 0.0847 0.0616 0.0279 0.0124
53
Table 3 Total Variation Distance for Frank (ρ = −0.64348) with m = 10
n 10 100 500 1000 2000 10000 50000
mean 0.8559 0.3316 0.1456 0.1029 0.0730 0.0324 0.0145
variance 0.00050 0.00072 0.00016 0.00008 0.00005 9 e-06 1 e-06
minimum 0.7966 0.2514 0.1019 0.0749 0.0505 0.0240 0.0107
maximum 0.9254 0.4144 0.1873 0.1325 0.0946 0.0433 0.0192
Table 4 Total Variation Distance for Gumbel (ρ = 0.84816) with m = 10
n 10 100 500 1000 2000 10000 50000
mean 0.7539 0.2658 0.1206 0.0854 0.0604 0.0268 0.0119
variance 0.00268 0.00085 0.00016 0.00007 0.00004 8 e-06 1 e-06
minimum 0.6556 0.1887 0.0841 0.0553 0.0407 0.0188 0.0080
maximum 0.9092 0.3548 0.1659 0.1141 0.0825 0.0363 0.0169
Table 5 Total Variation Distance for Plackett (ρ = −0.90005) with m = 10
n 10 100 500 1000 2000 10000 50000
mean 0.7110 0.2351 0.1105 0.0782 0.0552 0.0246 0.0109
variance 0.00509 0.00082 0.00016 0.00007 0.00004 8 e-06 1 e-06
minimum 0.5367 0.1460 0.0715 0.0528 0.0391 0.01644 0.0077
maximum 0.8907 0.03555 0.1542 0.1017 0.0896 0.0350 0.01570
Table 6 Total Variation Distance for Product d = 2 (ρ = 0) with m = 15
n 15 225 750 1500 3000 12000 60000
mean 0.9333 0.3421 0.2067 0.1453 0.1017 0.0509 0.0227
variance 0 0.00044 0.00011 0.00005 0.00002 7 e-06 1 e-06
minimum 0.9333 0.2755 0.1746 0.1253 0.0855 0.0415 0.0187
maximum 0.9333 0.4044 0.2382 0.1702 0.1180 0.0603 0.0274
Table 7 Total Variation Distance for Product d = 3 (ρ = 0) with m = 2
n 2 16 48 192 6144 24576 49152
mean 0.7500 0.1890 0.1150 0.0569 0.0101 0.0050 0.0035
variance 0 0.00567 0.00191 0.00045 0.00001 3 e-06 1 e-06
minimum 0.7500 0 0 0.0104 0.00162 0.0010 0.00038
maximum 0.7500 0.3750 0.29166 0.1718 0.0273 0.0112 0.0080
54
The sample d-copula of order m is already a d-copula which provides a quasi-nonparametric
method to estimate a d-copula C. Once m has been chosen, it becomes a nonparametric estimator
of the checkerboard approximation C(m), which is a good estimator of C, even for m relatively
small. After Cn
m has been constructed, since we know that its density is constant on each of the
d-boxes of the partition of [0, 1]d that it generates, it is absolutely trivial to generate random sam-
ples of size N ≥ 2 from it. We have written a program using the package R, that generates these
samples. Using this program; First, we generated a sample of size n, that we called original sample
size, denoted by OSS, from this sample we obtain the original modified sample generated from the
absolutely continuous copulas Clayton, AMH, Gumbel, Plackett, Normal, t-Student and Product,
respectively, using different values of Spearman’s rho . Second, for a given value of m, we ob-
tained Cn
m the sample copula of order m, corresponding to each of the original modified samples.
Third, using our program we generated a sample from the copula Cn
m in each case of size N, that
we called simulated sample size, denoted by SSS. In Figures 2.47 through Figure 2.53, we took
OSS = 5000, SSS = 5000 and m = 50, on the left hand side of each Figure we show the original
modified sample, and on the right hand side we show the simulated sample, as can be easily seen,
both samples are quite similar in all cases, which indicates that Cn
m , the sample copula of order 50
is a really good estimator of the original copula C.
In Figure 2.54, we took OSS = 20, m = 20, that is, the case m = n, SSS = 1000, and the sample
was taken from Π2 the product copula, in this case the original modified sample on the left hand
side looks like an independent sample, but the simulated sample on the right hand side does not
look at all, like an independent sample on [0, 1]2. In this case the sample copula C20
20
corresponds
to the linear B-spline given in [43], which generates poor samples for the independence copula Π2.
In Figures 2.55, 2.56 2.57 and 2.58, we sample from M2 , Example 3.3, Example 3.4 and Example
3.5 in Nelsens book [37]. We took OSS = 5000, SSS = 5000 and m = 50, except on Figure 2.58,
in which m = 100. As we can see in all these four Figures the simulated samples replicate the real
supports of these four singular copulas. The only difference is that the supports of the simulated
samples are slightly enlarged versions (mounted on little squares) of the real support, due to the
definition of the sample copula of order m. However, from looking at these simulated samples, it
is obvious, that we can deduce if the original copulas are singular.
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0.0 0.2 0.4 0.6 0.8 1.0
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Simulated Sample
Clayton Copula
Figure 2.47: ρ = −0.7921, OSS=5000, SSS=5000, m = 50
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1.
0
Simulated Sample
AMH Copula
Figure 2.48: ρ = 0.3451, OSS=5000, SSS=5000, m = 50
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0.0 0.2 0.4 0.6 0.8 1.0
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Simulated Sample
Gumbel Copula
Figure 2.49: ρ = 0.4412, OSS=5000, SSS=5000, m = 50
56
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0.0 0.2 0.4 0.6 0.8 1.0
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0
Original Sample
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0.0 0.2 0.4 0.6 0.8 1.0
0.
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0.
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4
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6
0.
8
1.
0
Simulated Sample
Plackett Copula
Figure 2.50: ρ = −0.9262, OSS=5000, SSS=5000, m = 50
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0.0 0.2 0.4 0.6 0.8 1.0
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0.
8
1.
0
Simulated Sample
Normal Copula
Figure 2.51: ρ = 0.7341, OSS=5000, SSS=5000, m = 50
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0.0 0.2 0.4 0.6 0.8 1.0
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1.
0
Simulated Sample
T−Student Copula
Figure 2.52: ρ = 0.7341, OSS=5000, SSS=5000, m = 50
57
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Product Copula
Figure 2.53: ρ = 0, OSS=5000, SSS=5000, m = 50
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Product Copula
Figure 2.54: ρ = 0, OSS=20, SSS=1000, m = 20
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M Copula
Figure 2.55: ρ = 1, OSS=5000, SSS=5000, m = 50
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Example 3.3 Copula
Figure 2.56: ρ = 0, OSS=5000, SSS=5000, m = 50
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Example 3.4 Copula
Figure 2.57: ρ = 0.75, OSS=5000, SSS=5000, m = 50
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Example 3.5 Copula
Figure 2.58: ρ = 0.286, OSS=5000, SSS=5000, m = 100
59
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Mixture G−GIID−GIDI−GDII Copula
Figure 2.59: Dimension 3, model = 10, OSS=2000, SSS=2000, m = 50
Finally, in Figure 2.59 we sample in dimension d = 3 from model 10, which is a mixture of
Gumbel, GumbelIID, GumbelIDI and GumbelDII, and allows us to have different dependencies
on the vertices of the cube [0, 1]3 , with OSS = 2000, SSS = 2000 and m = 50, by its shape we
call this the pinhata copula, and both samples look alike. From all the Figures it is clear that the
sample copula of order m is a very nice estimator of the original copula C, when the sample size n
is not too small, and the value of the order m is not close to n. In general, we recommend to take
values of m ≤ n1/d.
60
3 Moments of some of the random variables associated with
the sample copula
In this section we show the characteristics of the distributions associated to the count of the boxes
generated by the uniform partition of order m of Ik = [0, 1]k.
We obtained similar results to the ones in [8], the identities corresponding to permutations and
combinations, used in this section, can be found in [27]. We use the following notation to indicate
the k-permutations of n
Pn
k =
n!
(n − k)!
.
To note that our case of study corresponds to observations of the modified sample from a d-copula
C or a continuous joint distribution H. Let Im = {1, · · · ,m}, if we consider the sample without
applying the rank transformation, then by [24] and [25] and considering m ≥ 2, d ≥ 2 and for
every i = (i1, · · · , id) ∈ Id
m
Rm
i =
〈
i1 − 1
m
,
i1
m
]
× · · · ×
〈
id − 1
m
,
id
m
]
the uniform partition of size m of Id = [0, 1]d, where the notation “〈” indicates “(” if i j > 1 and
“[” if i j = 1, for all j ∈ Im, the random vector
(
Ni | i = (i1, · · · , id) ∈ Id
m
)
have a multinomial distribution with parameters pi = VolC(Rm
i
), the C-volume of the region Ri,
for all i = (i1, · · · , id) ∈ Id
m; [25] affirms that it is only necessary to consider d(m − 1) + 1 of these
parameters as free, the remaining parameters are determined by the properties that satisfy a copula.
For the case of the original sample, in dimension d = 2, we have the following lemma (see [25],
for the proof of this result),
Lemma 3.1 Let m ≥ 2, we consider the uniform partition Ri j, i, j ∈ {1, · · · ,m}, of I2 = [0, 1]2, and
given a 2-copula C we define, for all i, j ∈ {1, · · · ,m}
pi j = VC(Ri j)
61
p11 p21 · · · pm1
...
...
· · · ...
p1m−1 p2m−1 · · · pmm−1
p1m p2m · · · pmm
0 1
m
2
m
m−1
m 1
1
m
2
m
m−1
m
1
Figure 3.1: Uniform partition in I2 = [0, 1]2
We have the following relations
0 ≤ ∑m−1
i=1 pi j ≤
1
m
( j ∈ {1, · · · ,m − 1})
0 ≤ ∑m−1
j=1 pi j ≤
1
m
(i ∈ {1, · · · ,m − 1})
1 − 2
m
≤
m−1
∑
i, j=1
pi j ≤ 1 − 1
m
for i, j ∈ {1, · · · ,m − 1}
pim =
1
m
−
m−1
∑
j=1
pi j, pm j =
1
m
−
m−1
∑
i=1
pi j
and
pmm =
1
m
−
m−1
∑
j=1
pm j =
1
m
−
m−1
∑
i=1
pim
Let n,m ≥ 2, where m divides n, and Ri j, i, j ∈ {1, · · · ,m}, the uniform partition of I2 = [0, 1]2.
Let Ni j, i, j ∈ {1, · · · ,m}, be the random variables that indicates the number of observation in the
regions Ri j, i, j ∈ {1, · · · ,m}, respectively, then the distribution of the random vector
(Ni j, i, j ∈ {1, · · · ,m})
is multinomial with parameters n and pi j, i, j ∈ {1, · · · ,m}. It is important to note that only (m−1)2
values of the probabilities pi j, i, j ∈ {1, · · · ,m− 1}, are free, the remaining probabilities pi j, can be
written in terms of the previous probabilities (in the Figure 3.1 they are appear in red).
62
In dimension d = 3 we have the following lemma (in [25], we found the proof of this result),
Lemma 3.2 Let m ≥ 2, we consider the uniform partition Ri jk, i, j, k ∈ {1, · · · ,m}, of the unit cube
I3 = [0, 1]3, given a 3-copula C we define
pi jk = VC(Ri jk)
p111 p211 · · · pm11
p112 p212 · · · pm12
.
.
.
.
.
.
· · · .
.
.
p11m p21m · · · pm1m
pm
21
· · ·
pm
m
1
pm
22
· · ·
pm
m
2.
.
.
· · ·
.
.
.
pm
2m
· · ·
pm
m
m
p12m p22m · · ·
· · · · · · · · ·
p1mm p2mm · · ·
0 1
m
2
m
m−1
m 1
1
m
2
m
m−1
m
1
1
m
2
m
m−1
m
1
Figure 3.2: Uniform partition in I3 = [0, 1]3
We have the following relations,
Let i ∈ {1, · · · ,m − 1}
0 ≤
∑
( j,k)∈{1,···,m}2\{(m,m)}
pi jk ≤
1
m
.
Let j ∈ {1, · · · ,m − 1}
0 ≤
∑
(i,k)∈{1,···,m}2\{(m,m)}
pi jk ≤
1
m
.
Let k ∈ {1, · · · ,m − 1}
0 ≤
∑
(i, j)∈{1,···,m}2\{(m,m)}
pi jk ≤
1
m
.
We define
Id
m,r = {(i1, · · · , id) ∈ {1, · · · ,m}d | (d − r) coordinates equal to m}
also it holds
2 − 3
m
≤
3
∑
r=2










(r − 1)
∑
(i, j,k)∈I3
m,r
pi jk










≤ 2 − 2
m
.
63
We observe that the probabilities
p1mm, · · · , pm−1mm, pm1m, · · · , pmm−1m, pmm1, · · · , pmmm−1 y pmmm
can be written in terms of the other probabilities pi jk, for example
p1mm =
1
m
−
∑
( j,k)∈{1,···,m}2\{(m,m)}
p1 jk y pmmm =
3
m
− 2 +
3
∑
r=2










(r − 1)
∑
(i, j,k)∈I3
m,r
pi jk










.
Similarly to the case d = 2, let n,m ≥ 2, where m divides n, and Ri jk, i, j, k ∈ {1, · · · ,m}, the regions
corresponding to the uniform partition of the unit cube I3 = [0, 1]3. Let Ni jk, i, j, k ∈ {1, · · · ,m},
be the random variables that indicates the number of observations in the regions Ri jk, i, j, k ∈
{1, · · · ,m}, respectively, then the distribution of de random vector
(Ni jk, i, j, k ∈ {1, · · · ,m})
is multinomial with parameters n and pi jk, i, j, k ∈ {1, · · · ,m}. Regarding to the probabilities pi jk,
i, j, k ∈ {1, · · · ,m} is only necessary to consider m3 − (3m − 2) free parameters (in the Figure 3.2
they are marked with red color the regions determined for the remaining regions).
The distributions described in the above lemmas corresponds to the case where we considered the
original sample, that is, the sample without the rank transformation. Following we present the
procedure to make the count of the observations in the boxes considering the modified sample.
Remark 3.3 In the grid of I2 = [0, 1]2, generated by the partitions Px = {0, 1/n, · · · , (n − 1)/n, 1},
in the first coordinate X and Py = {0, 1/n, · · · , (n − 1)/n, 1} in the second coordinate Y, there exist
n! different ways in which can be observed n rank statistics from a sample of size n, because points
are not considered if they have the same value in some coordinate.
Figure 3.3: Grid in I2 = [0, 1]2
In the Figure 3.3 we show, in red, a point that should not be considered, because it has the same
second coordinate as another point (marked in blue).
64
We consider n ≥ m ≥ 2, where m divides n, and l = n/m. let R11 = [0, l/n]2 and let N11 be the
random variable that indicates the number of the observations k in R11, with k ∈ {0, 1, · · · , l} (see
Figure 3.4).
R11
0 1
m
1
m
Figure 3.4: Region R11
To calculate P{N11 = k}, with k ∈ {0, 1, · · · , l}, we first select the number of ways in which we can
select the coordinates in the first coordinate X, from these k observations, this count corresponds
to
(
l
k
)
(see Figure 3.5).
R11
We select k posibilities
(marked in red) among the l
total options
regardless of the order
0 1
m
1
m
Figure 3.5
subsequently we calculate the number of ways in which we can select the coordinates in the second
coordinate Y from these k observations, this count is given by Pl
k
(see Figure 3.6).
65
R11
We select k posibilities
(marked in red) among the l
total options
considering the order
0 1
m
1
m
Figure 3.6
Let R̂12 = [0, l/n]× [l/n, 1], the coordinates in the first coordinate X of the l− k points that we must
see in the region R̂12 can be selected of
(
l−k
l−k
)
= 1 way, the coordinates in the second coordinate Y,
can be selected of Pn−l
l−k
different ways (see Figure 3.7).
R11
R̂12
0 1
m
1
m
Figure 3.7
Let R̂22 = [l/n, 1]× [0, 1], the n− l points that we must see in this region, they have (n− l)! different
ways to appear (see Figure 3.8).
R11
R̂12 R̂22
0 1
m
1
m
Figure 3.8
66
From this observations, the number in which k points can be observed in the region R11 = [0, l/n]2,
l− k points in the region R̂12 = [0, l/n]× (l/n, 1] and n− l points in the region R̂22 = (l/n, 1]× [0, 1],
is
(
l
k
)
Pl
k
(
l − k
l − k
)
Pn−l
l−k(n − l)!
from n! possibilities.
The counting procedure described above, it can be generalized to higher dimensions, considering
permutations in the counting process of the different coordinates respect to the first coordinate axis.
3.1 Case: dimension two when m divides n
In this part, we present the distribution and moments of the random variables associated to the
counting in the boxes generated by the uniform partition of size m, with m ≥ 2, of I2 = [0, 1]2,
induced by the modified sample from the product copula.
Definition 3.4 Let m ≥ 2, n ∈ N, where m divides n, and l = n/m. We define the following regions
in the unit square I2 = [0, 1]2
R11 = [0, l/n] × [0, l/n]
R1 = [0, l/n] × (l/n, 1]
R = (l/n, 1] × [0, 1] .
Remark 3.5 In the following results we consider that in the unit square I2 exists n points corres-
ponding to the rank transformation of a sample of size n from the product copula.
The following results describe the probability distribution of the random variables associated with
the counting of observations in the boxes generated by the uniform partition.
Lemma 3.6 Let k1 be the number of points in the region R11 and x1 the number of points in the
region R1, we have that
∑
k1+x1=l
(
l
k1
)
Pl
k1
(
l − k1
x1
)
Pn−l
x1
= Pn
l .
Proof: We can observe that
(
l − k1
x1
)
=
(
x1
x1
)
= 1
67
then
∑
k1+x1=l
(
l
k1
)
Pl
k1
(
l − k1
x1
)
Pn−l
x1
=
∑
k1+x1=l
(
l
k1
)
Pl
k1
Pn−l
x1
=
∑
k1+x1=l
l!
k1!(l − k1)!
l!
(l − k1)!
(n − l)!
(n − l − x1)!
=
∑
k1+x1=l
l!
k1!(l − k1)!
l!
x1!
(n − l)!
(n − l − x1)!
= l!
∑
k1+x1=l
(
l
k1
)(
n − l
x1
)
= l!
(
n
l
)
(Vandermonde’s identity)
= Pn
l .

Lemma 3.7 Let N11 be the random variable that indicates the number of observations falling in
the region R11, then the following equality holds
P{N11 = k1} =
l!(n − l)!
n!
(
l
k1
)(
n − l
l − k1
)
=
(
l
k1
)(
n−l
l−k1
)
(
n
l
)
and the random variable N11 has a hypergeometric distribution con parameters: n the population
size, l the class size and l the sample size.
Proof: From the proof of Lemma 3.6, we have
∑
k1+x1=l
(
l
k1
)
Pl
k1
(
l − k1
x1
)
Pn−l
x1
= l!
∑
k1+x1=l
(
l
k1
)(
n − l
x1
)
,
this equality indicates the number of ways in which we can have k1 + x1 points in the region
R11 ∪ R1; fixing the value k1, this result is multiplied by (n − l)! (number of ways that we can have
n − l points in the region R discarding l possibilities corresponding to the coordinates occupied by
the observations in the region R11 ∪ R1) and divided by n!, the total number of ways that we can
observe n points in I2 = [0, 1]2. 
68
Then
l
∑
k1=1
P{N11 = k1} =
l
∑
k1=1
l!(n − l)!
n!
(
l
k1
)(
n − l
l − k1
)
=
(n − l)!
n!
l!
l
∑
k1=1
(
l
k1
)(
n − l
l − k1
)
=
(n − l)!
n!
Pn
l
= 1.
Theorem 3.8 Let n and m be integers greater than or equal to two, where m divides n, and l = n/m.
Let N11 be the random variable indicating the number of observations falling in R11, when we take
a sample of size n from the product copula, then
E(N11/n) =
1
m2
, E((N11/n)2) =
l2(l − 1)2
n3(n − 1)
+
1
nm2
and
Var(N11/n) =
l2(l − 1)2
n3(n − 1)
+
1
nm2
−
(
1
m2
)2
.
Proof: From the Lemma 3.7, we have
E(N11) =
l
∑
k1=0
k1
l!(n − l)!
n!
(
l
k1
)(
n − l
l − k1
)
=
l!l(n − l)!
n!
l
∑
k1=1
(
l − 1
k1 − 1
)(
n − l
l − k1
)
=
l!l(n − l)!
n!
l−1
∑
u1=0
(
l − 1
u1
)(
n − l
l − 1 − u1
)
(u1 = k1 − 1)
=
l!l(n − l)!
n!
(
n − 1
l − 1
)
(Vandermonde’s identity)
=
l2
n
=
n
m2
and
E(N11/n) =
1
m2
.
69
Then,
E((N11)2) =
l
∑
k1=0
k2
1
l!(n − l)!
n!
(
l
k1
)(
n − l
l − k1
)
=
l
∑
k1=0
(k1(k1 − 1) + k1)
l!(n − l)!
n!
(
l
k1
)(
n − l
l − k1
)
=
l
∑
k1=2
k1(k1 − 1)
l!(n − l)!
n!
(
l
k1
)(
n − l
l − k1
)
+
l
∑
k1=0
k1
l!(n − l)!
n!
(
l
k1
)(
n − l
l − k1
)
=
l
∑
k1=2
l!(n − l)!
n!
l(l − 1)
(
l − 2
k1 − 2
)(
n − l
l − k1
)
+
n
m2
=
l!(n − l)!
n!
l(l − 1)
l−2
∑
u1=0
(
l − 2
u1
)(
n − l
(l − 2) − u1
)
+
n
m2
(u1 = k1 − 2)
=
l!(n − l)!
n!
l(l − 1)
(
n − 2
l − 2
)
+
n
m2
=
l2(l − 1)2
n(n − 1)
+
n
m2
.
Therefore
E((N11/n)2) =
l2(l − 1)2
n3(n − 1)
+
1
nm2
and
Var(N11/n) = E((N11/n)2 − (E(N11/n))2
=
l2(l − 1)2
n3(n − 1)
+
1
nm2
−
(
1
m2
)2
.

Remark 3.9 To evaluate the covariance, we consider two cases, corresponding to the position of
the boxes in the square I2 = [0, 1]2, as illustrated in Figure 3.9.
70
R11
R12
R11
R22
0 1
m
1
m
2
m
0 1
m
2
m
1
m
2
m
Figure 3.9: Covariances cases d = 2.
The results presented below correspond to the calculation of the covariances for the first case.
Definition 3.10 Let r ∈ {2, · · · ,m}, we define the following region in the unit square I2 = [0, 1]2
R1r = [0, l/n] × ((l/n)(r − 1), (l/n)r] .
Lemma 3.11 Let k1 be the number of points in R11, let k2 be the number of points in R1r, r ∈
{2, · · · ,m}, and let x1 be the number of points in ([0, l/n] × [0, 1])\(R11 ∪ R1r), then
∑
k1+k2+x1=l
(
l
k1
)
Pl
k1
(
l − k1
k2
)
Pl
k2
(
l − k1 − k2
x1
)
Pn−2l
x1
= Pn
l .
Proof: We have that
(
l − k1 − k2
x1
)
=
(
x1
x1
)
= 1
and
∑
k1+k2+x1=l
(
l
k1
)
Pl
k1
(
l − k1
k2
)
Pl
k2
(
l − k1 − k2
x1
)
Pn−2l
x1
=
∑
k1+k2+x1=l
(
l
k1
)
Pl
k1
(
l − k1
k2
)
Pl
k2
Pn−2l
x1
=
∑
k1+k2+x1=l
l!
k1!(l − k1)!
l!
(l − k1)!
(l − k1)!
k2!(l − k1 − k2)!
· l!
(l − k2)!
(n − 2l)!
(n − 2l − x1)!
= l!
∑
k1+k2+x1=l
(
l
k1
)(
l
k2
)(
n − 2l
x1
)
= l!
(
n
l
)
= Pn
l .
71

Lemma 3.12 Let N11 be the random variable indicating the number of observations falling in R11
and let N1r, r ∈ {2, · · · ,m}, be the random variable indicating the number of observations falling
in R1r then
P{N11 = k1,N1r = k2} =
l!(n − l)!
n!
(
l
k1
)(
l
k2
)(
n − 2l
l − k1 − k2
)
.
Proof: We consider
l!
∑
k1+k2+x1=l
(
l
k1
)(
l
k2
)(
n − 2l
x1
)
= Pn
l
from the proof of Lemma 3.11, using the notation of the Definition 3.4, this equality indicates the
number of ways in which we can have k1 + k2 + x1 points in the region R11 ∪ R1; similarly to the
Lemma 1.4, setting the values k1 and k2, this result is multiplied by (n − l)! (number of ways that
we can have n − l points in the region R discarding l possibilities corresponding to the coordinates
occupied by the observations in the region R11 ∪ R1) and divided by n!, the total number of ways
that we can observe n points in I2 = [0, 1]2. 
We can observe that
∑
k1+k2=l
P{N11 = k1,N1r = k2} =
∑
k1+k2=l
l!(n − l)!
n!
(
l
k1
)(
l
k2
)(
n − 2l
l − k1 − k2
)
=
(n − l)!
n!
l!
∑
k1+k2+x1=l
(
l
k1
)(
l
k2
)(
n − 2l
x1
)
=
(n − l)!
n!
Pn
l
= 1.
Theorem 3.13 Let n and m be integers greater than or equal to two, where m divides n, and
l = n/m. Let N11 be the random variable indicating the number of observations falling in R11
(Definition 3.4) and let N1r, r ∈ {2, · · · ,m}, be the random variable indicating the number of
observations falling in R1r (Definition 3.10), when we consider a sample of size n from the product
copula, then
Cov(N11/n,N1r/n) =
l3(l − 1)
n3(n − 1)
−
(
1
m2
)2
.
72
Proof:
E(N11N1r) =
∑
k1+k2=l
k1k2
l!(n − l)!
n!
(
l
k1
)(
l
k2
)(
n − 2l
l − k1 − k2
)
=
l!(n − l)!
n!
∑
k1+k2+x1=l
k1k2
(
l
k1
)(
l
k2
)(
n − 2l
x1
)
=
l!(n − l)!
n!
l−2
∑
x1=0
(
n − 2l
x1
)
∑
k1+k2=l−x1
k1k2
(
l
k1
)(
l
k2
)
=
l!(n − l)!l2
n!
l−2
∑
x1=0
(
n − 2l
x1
)
∑
(k1−1)+(k2−1)=(l−2)−x1
(
l − 1
k1 − 1
)(
l − 1
k2 − 1
)
=
l!(n − l)!l2
n!
l−2
∑
x1=0
(
n − 2l
x1
)(
2l − 2
(l − 2) − x1
)
=
l!(n − l)!l2
n!
(
n − 2
l − 2
)
=
l!l2(n − l)!
n!
(n − 2)!
(l − 2)!(n − l)!
=
l3(l − 1)
n(n − 1)
and
Cov(N11/n,N1r/n) = E((N11/n)(N1r/n)) − E(N11/n)E(N1r/n)
=
l3(l − 1)
n3(n − 1)
−
(
1
m2
)2
.

The next results correspond to the calculation of the covariances in the second case.
Lemma 3.14 Let k1, x1 and x2 be the number of observed points in the regions R11 (Definition
3.4), R12 = [0, l/n] × (l/n, 2l/n] and R∗
13
= [0, l/n] × (2l/n, 1], respectively, let y1, k2 and y2 be
the number of observed points in R21 = (l/n, 2l/n] × [0, l/n], R22 = (l/n, 2l/n] × (l/n, 2l/n] and
R∗
23
= (l/n, 2l/n] × (2l/n, 1], respectively, then
∑
k1+x1+x2=l
∑
k2+y1+y2=l
(
l
k1
)
Pl
k1
(
l − k1
x1
)
Pl
x1
(
l − k1 − x1
x2
)
Pn−2l
x2
73
·
(
l
y1
)
Pl−k1
y1
(
l − y1
k2
)
P
l−x1
k2
(
l − y1 − k2
y2
)
Pn−2l−x2
y2
= Pn−l
l Pn
l .
Proof: We have that
(
l − k1 − x1
x2
)
= 1,
(
l − y1 − k2
y2
)
= 1
and
∑
k1+x1+x2=l
∑
k2+y1+y2=l
(
l
k1
)
Pl
k1
(
l − k1
x1
)
Pl
x1
Pn−2l
x2
(
l
y1
)
Pl−k1
y1
(
l − y1
k2
)
P
l−x1
k2
Pn−2l−x2
y2
= (l!)2
∑
k1+x1+x2=l
∑
k2+y1+y2=l
(
l
k1
)(
l
x1
)(
n − 2l
x2
)(
l − k1
y1
)(
l − x1
k2
)(
n − 2l − x2
y2
)
= (l!)2
∑
k1+x1+x2=l
(
l
k1
)(
l
x1
)(
n − 2l
x2
)(
n − k1 − x1 − x2
l
)
= (l!)2
(
n − l
l
)
∑
k1+x1+x2=l
(
l
k1
)(
l
x1
)(
n − 2l
x2
)
= (l!)2
(
n − l
l
)(
n
l
)
= Pn−l
l Pn
l .

Lemma 3.15 Let N11 be the random variable indicating the number of observations falling in
R11 (Definition 3.4) and N22 the random variable indicating the number of observations falling in
R22 = (l/n, 2l/n] × (l/n, 2l/n] then
P{N11 = k1,N22 = k2} =
(l!)2(n − 2l)!
n!
∑
x1+x2=l−k1
∑
y1+y2=l−k2
(
l
k1
)(
l
x1
)(
n − 2l
x2
)(
l − k1
y1
)(
l − x1
k2
)(
n − 2l − x2
y2
)
Proof: This equality is obtained from the proof Lemma 3.14, setting the values k1 and k2, and
multiplying by (n−2l)! (number of ways that we can have n−2l points in the region (2l/n, 1]×[0, 1]
discarding 2l possibilities corresponding to the coordinates occupied by the observations in the
region [0, 2l/n] × [0, 1]) and divided by n!, the total number of ways that we can observe n points
in I2 = [0, 1]2.
74
We can note that
l
∑
k1=0
l
∑
k2=0
P{N11 = k1,N22 = k2} =
(n − 2l)!
n!
Pn−l
l Pn
l
= 1.

Theorem 3.16 Let n and m be integers greater than or equal to two, where m divides n, and
l = n/m, with the same hypothesis of the Lemma 3.15 we have
Cov(N11/n,N22/n) =
l4
n3(n − 1)
− 1
m4
.
Proof:
E(N11N22) =
(l!)2(n − 2l)!
n!
∑
k1+x1+x2=l
k1
(
l
k1
)(
l
x1
)(
n − 2l
x2
)
∑
k2+y1+y2=l
k2
(
l − k1
y1
)(
l − x1
k2
)(
n − 2l − x2
y2
)
=
(l!)2l(n − 2l)!
n!
∑
(k1−1)+x1+x2=l−1
(
l − 1
k1 − 1
)(
l
x1
)(
n − 2l
x2
)
·(l − x1)
∑
(k2−1)+y1+y2=l−1
(
l − k1
y1
)(
l − x1 − 1
k2 − 1
)(
n − 2l − x2
y2
)
=
(l!)2l(n − 2l)!
n!
∑
(k1−1)+x1+x2=l−1
(
l − 1
k1 − 1
)(
l
x1
)(
n − 2l
x2
)
(l − x1)
(
n − 1 − l
l − 1
)
=
(l!)2l2(n − 2l)!
n!
(
n − 1 − l
l − 1
)
∑
(k1−1)+x1+x2=l−1
(
l − 1
k1 − 1
)(
l
x1
)(
n − 2l
x2
)
− (l!)2l2(n − 2l)!
n!
(
n − 1 − l
l − 1
)
∑
(k1−1)+(x1−1)+x2=l−2
(
l − 1
k1 − 1
)(
l − 1
x1 − 1
)(
n − 2l
x2
)
=
(l!)2l2(n − 2l)!
n!
(
n − 1 − l
l − 1
) [(
n − 1
l − 1
)
−
(
n − 2
l − 2
)]
=
(l!)2l2(n − 2l)!
n!
(n − 1 − l)!
(n − 2l)!(l − 1)!
(
(n − 1)!
(l − 1)!(n − l)!
− (n − 2)!
(l − 2)!(n − l)!
)
=
(l!)2l2(n − 1)!(n − 1 − l)!
n!((l − 1)!)2(n − l)!
− (l!)2l2(n − 2)!(n − 1 − l)!
n!(l − 1)!(l − 2)!(n − l)!
=
l4
n(n − l)
− l4(l − 1)
n(n − 1)(n − l)
75
=
l4
n(n − 1)
and
E((N11/n)(N22/n)) =
l4
n3(n − 1)
.
Therefore
Cov(N11/n,N22/n) = E((N11/n)(N22/n)) − E(N11/n)E(N22/n)
=
l4
n3(n − 1)
− 1
m4
.

We finalized this section with a theorem that indicates the joint probability distribution of the boxes
generated for the uniform partition of size m, with m ≥ 2, when m divides n.
Theorem 3.17 Let m ≥ 2, n ∈ N, where m divides n, l = n/m and Im = {1, · · ·m}, let Ri j, with
i, j ∈ Im, be the boxes of the uniform partition of size m of I2 = [0, 1]2 and let Ni j be the random
variables that indicate the number of observations falling in Ri j, respectively, for all i, j ∈ Im, when
we consider a sample of size n from the product copula; let ni j, with i, j ∈ Im, be zero or a positive
integer, satisfying the following restrictions
m
∑
j=1
ni j = l (for all i ∈ Im),
m
∑
i=1
ni j = l (for all j ∈ Im),
then
P







⋂
i, j∈Im
{Ni j = ni j}







=
(l!)2m
n!
∏
i j∈Im
ni j!
.
Proof: Proceeding as in Remark 3.3, first for the region R11, second for the region R12 and succes-
sively to the regions R13, · · · ,R1m,R21, · · · ,R2m,Rm1, · · · ,Rmm, we have that the number of ways in
which we can observe ni j points in each region Ri j, i, j ∈ Im, is given by
(
l
n11
)
Pl
n11
(
l − n11
n12
)
Pl
n12
· · ·
(
l −∑m−1
j=1 n1 j
n1m
)
Pl
n1m
·
(
l
n21
)
Pl−n11
n21
(
l − n21
n22
)
Pl−n12
n22
· · ·
(
l −∑m−1
j=1 n2 j
n2m
)
Pl−n1m
n2m
76
...
·
(
l
nm1
)
P
l−∑m−1
i=1 ni1
nm1
(
l − nm1
nm2
)
P
l−∑m−1
i=1 ni2
nm2
· · ·
(
l −∑m−1
j=1 nm j
nmm
)
P
l−∑m−1
i=1 nim
nmm
=
(l!)2m
∏
i, j∈Im
ni j!
,
finally, we divided between n!, the total number of possibilities in which we can see n points in
I2 = [0, 1]2, corresponding to the modified sample of the a sample of size n from the product
copula, therefore
P







⋂
i, j∈Im
{Ni j = ni j}







=
(l!)2m
n!
∏
i, j∈Im
ni j!
.

3.2 Case: dimension three when m divides n
In a similar way to the previous section, we present the distribution and moments of the random
variables associated to the counting in the boxes generated by the uniform partition of size m, with
m ≥ 2, of I3 = [0, 1]3, induced by the modified sample from the product copula.
Theorem 3.18 Let n and m be integers greater than or equal to two, where m divides n, and
l = n/m. Let N111 be the random variable that indicates the number of observations falling in
R111 = [0, l/n]3 when we consider a sample of size n from the product copula, then
E(N11/n) =
1
m3
, E((N11/n)2) =
l3(l − 1)3
n4(n − 1)2
+
1
nm3
and
Var(N111/n) =
l3(l − 1)3
n4(n − 1)2
+
1
nm3
−
(
1
m3
)2
.
Remark 3.19 Figure 3.10 shows the related boxes to the count of the number of observations
associated to the random variable N111, the Figure 3.11 show the region R = (l/n, 1]×[0, 1]×[0, 1],
in this region exist ((n − l)!)2 ways in which we can graph the transformed sample data, since that
l points are observed in the region R1 = [0, l/n] × [0, 1] × [0, 1].
77
R111
Figure 3.10: Box R111
R111
R
Figure 3.11: Box R
k
x1
x2
Figure 3.12: Box R1
Remark 3.20 The Figure 3.12 it shows the region R1 = [0, l/n]×[0, 1]×[0, 1], this region is divided
on three parts, R111 = [0, l/n]3, R112 = [0, l/n]×[0, l/n]×(l/n, 1] and R122 = [0, l/n]×(l/n, 1]×[0, 1],
the number of observations in the regions are denoted, respectively, by k, x1 and x2. The number
of ways in which we can select the value of k is
(
l
k
)
Pl
k
Pl
k
, the number of ways in which we can select
the value of x1, given the value k, is
(
l−k
x1
)
Pl−k
x1
Pn−l
x1
and the number of possibilities of the value x2 is,
given the values k and x1,
(
l−k−x1
x2
)
Pn−l
x2
P
n−k−x1
x2
.
Definition 3.21 Let k ∈ {0, 1, · · · , l}, we define
Gn,l
k
=
∑
x1+x2=l−k
(
l − k
x1
)
Pl−k
x1
Pn−l
x1
(
l − k − x1
x2
)
Pn−l
x2
Pn−k−x1
x2
=
l−k
∑
x1=0
(
l − k
x1
)
Pl−k
x1
Pn−l
x1
(
l − k − x1
l − k − x1
)
Pn−l
l−k−x1
P
n−k−x1
l−k−x1
=
l−k
∑
x1=0
(l − k)!
x1!(l − k − x1)!
(l − k)!
(l − k − x1)!
(n − l)!
(n − l − x1)!
(n − l)!
(n − 2l + k + x1)!
(n − k − x1)!
(n − l)!
= ((l − k)!)2
l−k
∑
x1=0
(
n − k − x1
l − k − x1
)(
n − l
l − k − x1
)(
n − l
x1
)
.
78
Remark 3.22 We have,
l
∑
k=0
(
l
k
)
Pl
kPl
kG
n,l
k
= ((Pn
l )!)2.
then
l
∑
k=0
(
l
k
)
(l!)2








l−k
∑
x1=0
(
n − k − x1
l − k − x1
)(
n − l
l − k − x1
)(
n − l
x1
)








((n − l)!)2 = (n!)2
Remark 3.23 Le k ∈ {0, 1, · · · , l}, we have
P{N = k} =
(
l
k
)
Pl
kPl
kG
n,l
k
((n − l)!)2/(n!)2
Using Remark 3.23, we can proof the Theorem 3.18,
E(N111) =
l
∑
k=0
k
(
l
k
)
(l!)2








l−k
∑
x1=0
(
n − k − x1
l − k − x1
)(
n − l
l − k − x1
)(
n − l
x1
)








((n − l)!)2/(n!)2
= l3
l
∑
k=1
(
l − 1
k − 1
)
((l − 1)!)2








(l−1)−(k−1)
∑
x1=0
(
(n − 1) − (k − 1) − x1
(l − 1) − (k − 1) − x1
)
(
(n − 1) − (l − 1)
(l − 1) − (k − 1) − x1
)(
(n − 1) − (l − 1)
x1
)]
(((n − 1) − (l − 1))!)2/(n!)2
= l3
l−1
∑
u=0
(
l − 1
u
)
((l − 1)!)2








(l−1)−u
∑
x1=0
(
(n − 1) − u − x1
(l − 1) − u − x1
)
(
(n − 1) − (l − 1)
(l − 1) − u − x1
)(
(n − 1) − (l − 1)
x1
)]
(((n − 1) − (l − 1))!)2/(n!)2
=
l3((n − 1)!)2
(n!)2
(Remark 3.22)
=
n3
m3
1
n2
=
n
m3
and
E
(
N111
n
)
=
1
m3
.
79
The second moment is obtained from the following equalities
E(N2
111) =
l
∑
k=0
k2
(
l
k
)
(l!)2








l−k
∑
x1=0
(
n − k − x1
l − k − x1
)(
n − l
l − k − x1
)(
n − l
x1
)








((n − l)!)2/(n!)2
=
l
∑
k=0
[k(k − 1) + k]
(
l
k
)
(l!)2








l−k
∑
x1=0
(
n − k − x1
l − k − x1
)(
n − l
l − k − x1
)(
n − l
x1
)








((n − l)!)2/(n!)2
=
l
∑
k=2
k(k − 1)
(
l
k
)
(l!)2








l−k
∑
x1=0
(
n − k − x1
l − k − x1
)(
n − l
l − k − x1
)(
n − l
x1
)








((n − l)!)2/(n!)2 +
n
m3
= (l(l − 1))3
l
∑
k=2
(
l − 2
k − 2
)
((l − 2)!)2








(l−2)−(k−2)
∑
x1=0
(
(n − 2) − (k − 2) − x1
(l − 2) − (k − 2) − x1
)
(
(n − 2) − (l − 2)
(l − 2) − (k − 2) − x1
)(
(n − 2) − (l − 2)
x1
)]
(((n − 2) − (l − 2))!)2/(n!)2 +
n
m3
= (l(l − 1))3
l−2
∑
u=0
(
l − 2
u
)
((l − 2)!)2








(l−2)−u
∑
x1=0
(
(n − 2) − u − x1
(l − 2) − u − x1
)
(
(n − 2) − (l − 2)
(l − 2) − u − x1
)(
(n − 2) − (l − 2)
x1
)]
(((n − 2) − (l − 2))!)2/(n!)2 +
n
m3
=
(l(l − 1))3((n − 2)!)2
(n!)2
+
n
m3
(Remark 3.22)
=
l3(l − 1)3
n2(n − 1)2
+
n
m3
.
and
E
(
(
N111
n
)2
)
=
l3(l − 1)3
n4(n − 1)2
+
1
nm3
.
Finally
Var(N111/n) = E((N111/n)2) − (E(N111/n))2
=
l3(l − 1)3
n4(n − 1)2
+
1
nm3
−
(
1
m3
)2
.

Remark 3.24 To evaluate the covariance, we consider three cases, with respect to the position
of the boxes R112 = [0, l/n] × [0, l/n] × (l/n, 2l/n], R122 = [0, l/n] × (l/n, 2l/n] × (l/n, 2l/n] and
R222 = (l/n, 2l/n]3, relatives to the box R111.
80
R111
R112
Figure 3.13: Case 1
R111
R222
Figure 3.14: Case 2
R111
R122
Figure 3.15: Case 3
Lemma 3.25 (Case 1) Let n and m be integers greater than or equal to two, where m divides n,
and l = n/m. Let k1 be the number of observations falling in R111 = [0, l/n]3, k2 the number of
observations falling in R112 = [0, l/n] × [0, l/n] × (l/n, 2l/n], x1 be the number of observations
falling in R113 = [0, l/n] × [0, l/n] × (l/n, 1] and x2 the number of observations falling in R∗ =
[0, l/n] × (l/n, 1] × [0, 1], when we considered a sample of size n from the product copula, then
∑
k1+k2+x1+x2=l
(
l
k1
)
Pl
k1
Pl
k1
(
l − k1
k2
)
P
l−k1
k2
Pl
k2
(
l − k1 − k2
x1
)
Pl−k1−k2
x1
Pn−2l
x1
·
(
l − k1 − k2 − x1
x2
)
Pn−l
x2
Pn−x1−k1−k2
x2
=
(
Pn
l
)2
.
Proof: We note
(
l − k1 − k2 − x1
x2
)
= 1
and
∑
k1+k2+x1+x2=l
(
l
k1
)
Pl
k1
Pl
k1
(
l − k1
k2
)
P
l−k1
k2
Pl
k2
(
l − k1 − k2
x1
)
Pl−k1−k2
x1
Pn−2l
x1
·
(
l − k1 − k2 − x1
x2
)
Pn−l
x2
Pn−x1−k1−k2
x2
81
=
∑
k1+k2+x1+x2=l
(
l
k1
)
Pl
k1
Pl
k1
(
l − k1
k2
)
P
l−k1
k2
Pl
k2
(
l − k1 − k2
x1
)
Pl−k1−k2
x1
Pn−2l
x1
Pn−l
x2
Pn−x1−k1−k2
x2
= (l!)2
∑
k1+k2+x1+x2=l
(
l
k1
)(
l
k2
)(
n − 2l
x1
)(
n − l
x2
)(
n − x1 − k1 − k2
x2
)
= (l!)2
∑
k1+k2+x1+x2=l
(
l
k1
)(
l
k2
)(
n − 2l
x1
)(
n − l
x2
)(
n − (l − x2)
x2
)
= (l!)2
l
∑
x2=0
(
n − l
x2
)(
n − (l − x2)
x2
)
∑
k1+k2+x1=l−x2
(
l
k1
)(
l
k2
)(
n − 2l
x1
)
= (l!)2
l
∑
x2=0
(
n − l
x2
)(
n − (l − x2)
x2
)(
n
l − x2
)
= (l!)2n!
l
∑
x2=0
1
x2!(n − l − x2)!
1
x2!(l − x2)!
=
(l!)2n!
l!(n − l)!
l
∑
x2=0
(
l
x2
)(
n − l
(n − l) − x2
)
=
(l!)2n!
l!(n − l)!
(
n
n − l
)
= (Pn
l )2.

Remark 3.26 Let n and m be integers greater than or equal to two, where m divides n, and l =
n/m. Let N111 be the random variable indicating the number of observations falling in R111 =
[0, l/n]3 and let N112 be the random variable indicating the number of observations falling in
R112 = [0, l/n] × [0, l/n] × (l/n, 2l/n], when we take a sample of size n from the product copula,
using the proof of the Lemma 3.25 we have
P{N111 = k1,N112 = k2} =
(l!)2((n − l)!)2
(n!)2
l−k1−k2
∑
x2=0
(
n − l
x2
)(
n − (l − x2)
x2
)(
l
k1
)(
l
k2
)(
n − 2l
l − x2 − k1 − k2
)
.
Theorem 3.27 With the same hypothesis of Remark 3.26, we have
Cov(N111/n,N112/n) =
l4(l − 1)2
n4(n − 1)2
−
(
1
m3
)2
.
82
Proof: From Lemma 3.25 and Remark 3.26 we have
E(N111N112) =
(l!)2((n − l)!)2
(n!)2
l−2
∑
x2=0
(
n − l
x2
)(
n − (l − x2)
x2
)
∑
k1+k2+x1=l−x2
k1k2
(
l
k1
)(
l
k2
)(
n − 2l
x1
)
=
l2(l!)2((n − l)!)2
(n!)2
l−2
∑
x2=0
(
n − l
x2
)(
n − (l − x2)
x2
)
·
∑
(k1−1)+(k2−1)+x1=(l−2)−x2
(
l − 1
k1 − 1
)(
l − 1
k2 − 1
)(
n − 2l
x1
)
=
l2(l!)2((n − l)!)2
(n!)2
l−2
∑
x2=0
(
n − l
x2
)(
n − (l − x2)
x2
)(
n − 2
(l − 2) − x2
)
=
l2(l!)2((n − l)!)2(n − 2)!
(n!)2(n − l)!(l − 2)!
l−2
∑
x2=0
(
n − l
x2
)(
l − 2
(l − 2) − x2
)
=
l2(l!)2((n − l)!)2(n − 2)!
(n!)2(n − l)!(l − 2)!
(
n − 2
l − 2
)
=
l4(l − 1)2
n2(n − 1)2
,
and
Cov(N111/n,N112/n) = E((N111/n)(N112/n)) − E(N111/n)E(N112/n)
=
l4(l − 1)2
n4(n − 1)2
−
(
1
m3
)2
.

Remark 3.28 (Case 2) Let n and m be integers greater than or equal to two, where m divides n,
and l = n/m, we consider the following regions
R111 = [0, l/n] × [0, l/n] × [0, l/n]
R112 = [0, l/n] × [0, l/n] × (l/n, 2l/n]
R113 = [0, l/n] × [0, l/n] × (2l/n, 1]
R121 = [0, l/n] × (l/n, 2l/n] × [0, l/n]
R122 = [0, l/n] × (l/n, 2l/n] × (l/n, 2l/n]
R123 = [0, l/n] × (l/n, 2l/n] × (2l/n, 1]
R131 = [0, l/n] × (2l/n, 1] × [0, l/n]
83
R132 = [0, l/n] × (2l/n, 1] × (l/n, 2l/n]
R133 = [0, l/n] × (2l/n, 1] × (2l/n, 1],
let x1, x2, x3, x4, x5, x6, x7, x8 and x9 be, respectively, the number of observations in the regions
R111,R112,R113,R121,R122,R123,R131,R132 and R133.
We define
x1 + x2 + x3 = A1
x4 + x5 + x6 = A2
x7 + x8 + x9 = A3
x1 + x4 + x7 = B1
x2 + x5 + x8 = B2
x3 + x6 + x9 = B3.
To count the number of ways in which we can select l from n points in the region R∗ = [0, l/n] ×
[0, 1] × [0, 1] we consider
1. Number of ways to select the first coordinate X
CX =
(
l
x1
)(
l − x1
x2
)(
l − x1 − x2
x3
)(
l − x1 − x2 − x3
x4
)(
l − x1 − x2 − x3 − x4
x5
)
·
(
l − x1 − x2 − x3 − x4 − x5
x6
)(
l − x1 − x2 − x3 − x4 − x5 − x6
x7
)
·
(
l − x1 − x2 − x3 − x4 − x5 − x6 − x7
x8
)(
l − x1 − x2 − x3 − x4 − x5 − x6 − x7 − x8
x9
)
= l!
1
x1!
1
x2!
1
x3!
1
x4!
1
x5!
1
x6!
1
x7!
1
x8!
1
x9!
.
2. Number of ways to select the second coordinate Y
CY = Pl
x1
Pl−x1
x2
Pl−x1−x2
x3
Pl
x4
Pl−x4
x5
Pl−x4−x5
x6
Pn−2l
x7
Pn−2l−x7
x8
Pn−2l−x7−x8
x9
=
l!
(l − A1)!
l!
(l − A2)!
(n − 2l)!
(n − 2l − A3)!
.
3. Number of ways to select the third coordinate Z
CZ = Pl
x1
Pl
x2
Pn−2l
x3
Pl−x1
x4
Pl−x2
x5
Pn−2l−x3
x6
Pl−x1−x4
x7
Pl−x2−x5
x8
Pn−2l−x3−x6
x9
.
84
Lemma 3.29 Using the notation of Remark 3.28 we have
∑
A1+A2+A3=l
∑
x1+x2+x3=A1
∑
x4+x5+x6=A2
∑
x7+x8+x9=A3
CX ·CY ·CZ = (Pn
l )2.
Proof: We have the following identities
∑
A1+A2+A3=l
∑
x1+x2+x3=A1
∑
x4+x5+x6=A2
∑
x7+x8+x9=A3
CX ·CY ·CZ
=
∑
A1+A2+A3=l
∑
x1+x2+x3=A1
∑
x4+x5+x6=A2
∑
x7+x8+x9=A3
l!
l!
(l − A1)!
l!
(l − A2)!
(n − 2l)!
(n − 2l − A3)!
·
(
l
x1
)(
l
x2
)(
n − 2l
x3
)(
l − x1
x4
)(
l − x2
x5
)(
n − 2l − x3
x6
)(
l − x1 − x4
x7
)(
l − x2 − x5
x8
)(
n − 2l − x3 − x6
x9
)
=
∑
A1+A2+A3=l
l!
l!
(l − A1)!
l!
(l − A2)!
(n − 2l)!
(n − 2l − A3)!
(
n
A1
)(
n − A1
A2
)(
n − A1 − A2
A3
)
=
l!n!
(n − l)!
∑
A1+A2+A3=l
(
l
A1
)(
l
A2
)(
n − 2l
A3
)
=
l!n!
(n − l)!
(
n
l
)
= (Pn
l )2.

Remark 3.30 Let n and m be integers greater than or equal to two, where m divides n, and l =
n/m, we consider the following regions
R211 = (l/n, 2l/n] × [0, l/n] × [0, l/n]
R212 = (l/n, 2l/n] × [0, l/n] × (l/n, 2l/n]
R213 = (l/n, 2l/n] × [0, l/n] × (2l/n, 1]
R221 = (l/n, 2l/n] × (l/n, 2l/n] × [0, l/n]
R222 = (l/n, 2l/n] × (l/n, 2l/n] × (l/n, 2l/n]
R223 = (l/n, 2l/n] × (l/n, 2l/n] × (2l/n, 1]
R231 = (l/n, 2l/n] × (2l/n, 1] × [0, l/n]
R232 = (l/n, 2l/n] × (2l/n, 1] × (l/n, 2l/n]
R233 = [l/n, 2l/n] × (2l/n, 1] × (2l/n, 1],
85
let y1, y2, y3, y4, y5, y6, y7, y8 and y9 be, respectively, the number of observations in the regions
R211,R212,R213,R221,R222,R223,R231,R232 and R233.
We define
y1 + y2 + y3 = Â1
y4 + y5 + y6 = Â2
y7 + y8 + y9 = Â3
y1 + y4 + y7 = B̂1
y2 + y5 + y8 = B̂2
y3 + y6 + y9 = B̂3.
To count the number of ways in which we can select l from n − l points in the region R∗∗ =
(l/n, 2l/n] × [0, 1] × [0, 1] we consider
1. Number of ways to select the first coordinate X
CX =
(
l
y1
)(
l − y1
y2
)(
l − y1 − y2
y3
)(
l − y1 − y2 − y3
y4
)(
l − y1 − y2 − y3
y5
)
·
(
l − y1 − y2 − y3 − y4 − y5
y6
)(
l − y1 − y2 − y3 − y4 − y5 − y6
y7
)
·
(
l − y1 − y2 − y3 − y4 − y5 − y6 − y7
y8
)(
l − y1 − y2 − y3 − y4 − y5 − y6 − y7 − y8
y9
)
= l!
1
y1!
1
y2!
1
y3!
1
y4!
1
y5!
1
y6!
1
y7!
1
y8!
1
y9!
.
2. Number of ways to select the second coordinate Y (we consider the notation of Remark 3.28)
CY = Pl−A2
y5
Pl−A2−y5
y4
Pl−A2−y4−y5
y6
Pl−A1
y2
Pl−A1−y2
y1
Pl−A1−y2−y1
y3
Pn−2l−A3
y8
Pn−2l−A3−y8
y7
Pn−2l−A3−y8−y7
y9
.
3. Number of ways to select the third coordinate Z (we consider the notation of Remark 3.28)
CZ = l−B1 Py1
l−B1−y1 Py4
l−B1−y1−y4 Py7
l−B2 Py2
l−B2−y2 Py5
l−B2−y2−y5 Py8
n−2l−B3 Py3
n−2l−B3−y3 Py6
· n−2l−B3−y3−y6 Py9
=
(l − B1)!
(l − B1 − B̂1)!
(l − B2)!
(l − B2 − B̂2)!
(n − 2l − B3)!
(n − 2l − B3 − B̂3)!
.
86
Lemma 3.31 Using the notation of Remark 3.30 and Remark 3.28, we have that
∑
B̂1+B̂2+B̂3=l
∑
y2+y5+y8=B̂2
∑
y1+y4+y7=B̂1
∑
y3+y6+y9=B̂3
CX ·CY ·CZ = (Pn−l
l )2.
Proof: We have the following identities
∑
B̂1+B̂2+B̂3=l
∑
y2+y5+y8=B̂2
∑
y1+y4+y7=B̂1
∑
y3+y6+y9=B̂3
CX ·CY ·CZ
=
∑
B̂1+B̂2+B̂3=l
∑
y2+y5+y8=B̂2
∑
y1+y4+y7=B̂1
∑
y3+y6+y9=B̂3
l!
(l − B1)!
(l − B1 − B̂1)!
(l − B2)!
(l − B2 − B̂2)!
(n − 2l − B3)!
(n − 2l − B3 − B̂3)!
·
(
l − A2
y5
)(
l − A1
y2
)(
n − 2l − A3
y8
)(
l − A2 − y5
y4
)(
l − A1 − y2
y1
)(
n − 2l − A3 − y8
y7
)
·
(
l − A2 − y4 − y5
y6
)(
l − A1 − y2 − y1
y3
)(
n − 2l − A3 − y8 − y7
y9
)
=
∑
B̂1+B̂2+B̂3=l
l!
(l − B1)!
(l − B1 − B̂1)!
(l − B2)!
(l − B2 − B̂2)!
(n − 2l − B3)!
(n − 2l − B3 − B̂3)!
(
n − l
B̂2
)
·
(
n − l − B̂2
B̂1
)(
n − l − B̂1 − B̂2
B̂3
)
=
∑
B̂1+B̂2+B̂3=l
l!
(n − l)!
(n − 2l)!
(
l − B1
B̂1
)(
l − B2
B̂2
)(
n − 2l − B3
B̂3
)
= l!
(n − l)!
(n − 2l)!
(
n − l
l
)
= (Pn−l
l )2.

Remark 3.32 We use the notation on Remark 3.28 and Remark 3.30, and the results in the proofs
of the Lemma 3.29 and the Lemma 3.31. Let n and m be integers greater than or equal to two, where
m divides n, and l = n/m. Let N111 be the random variable indicating the number of observations
falling in R111 and N222 the random variable indicating the number of observations falling in R222,
when we considered a sample of size n from the product copula, then
P (N111 = x1,N222 = y5) =








∑
A1+A2+A3=l
∑
x2+x3=A1−x1
∑
x4+x5+x6=A2
∑
x7+x8+x9=A3
l!
l!
(l − A1)!
l!
(l − A2)!
(n − 2l)!
(n − 2l − A3)!
(
l
x1
)(
l
x2
)(
n − 2l
x3
)(
l − x1
x4
)(
l − x2
x5
)(
n − 2l − x3
x6
)
87
(
l − x1 − x4
x7
)(
l − x2 − x5
x8
)(
n − 2l − x3 − x6
x9
)]
·









∑
B̂1+B̂2+B̂3=l
∑
y2+y8=B̂2−y5
l!
(l − B1)!
(l − B1 − B̂1)!
(l − B2)!
(l − B2 − B̂2)!
(n − 2l − B3)!
(n − 2l − B3 − B̂3)!
(
l − A2
y5
)(
l − A1
y2
)
(
n − 2l − A3
y8
)(
n − l − B̂2
B̂1
)(
n − l − B̂1 − B̂2
B̂3
)]
·
[
((n − 2l)!)2
(n!)2
]
.
Remark 3.33 If we use the notation on Remark 3.28 and Remark 3.30, then
∑
B̂1+B̂2+B̂3=l
∑
y2+y8+(y5−1)=B̂2−1
l!
(l − B1)!
(l − B1 − B̂1)!
(l − B2)!
(l − B2 − B̂2)!
(n − 2l − B3)!
(n − 2l − B3 − B̂3)!
(
l − A2 − 1
y5 − 1
)
(
l − A1
y2
)(
n − 2l − A3
y8
)(
n − l − B̂2
B̂1
)(
n − l − B̂1 − B̂2
B̂3
)
=
∑
B̂1+B̂2+B̂3=l
l!
(l − B1)!
(l − B1 − B̂1)!
(l − B2)!
(l − B2 − B̂2)!
(n − 2l − B3)!
(n − 2l − B3 − B̂3)!
(
n − l − 1
B̂2 − 1
)(
n − l − B̂2
B̂1
)
(
n − l − B̂1 − B̂2
B̂3
)
=
∑
B̂1+B̂2+B̂3=l
l!
(l − B1)!
(l − B1 − B̂1)!
(l − B2)!
(l − B2 − B̂2)!
(n − 2l − B3)!
(n − 2l − B3 − B̂3)!
(n − l − 1)!
(B̂2 − 1)!
1
B̂1!
1
B̂3!
1
(n − 2l)!
=
l!(n − l − 1)!
(n − 2l)!
(l − B2)
∑
B̂1−1+B̂2+B̂3=l−1
(
l − B1
B̂1
)(
l − B2 − 1
B̂2 − 1
)(
n − 2l − B3
B̂3
)
=
l!(n − l − 1)!
(n − 2l)!
(l − B2)
(
n − l − 1
l − 1
)
.
Remark 3.34 Using the notation on Remark 3.28 and Remark 3.30, we have
(
l
x2
)(
l − x2
x5
)(
l − x2 − x5
x8
)
=
l!
x2!(l − x2)!
(l − x2)!
x5!(l − x2 − x5)!
(l − x2 − x5)!
x8!(l − x2 − x5 − x8)!
=
l
l − x2
(l − 1)!
x2!(l − x2 − 1)!
l − x2
l − x2 − x5
(l − x2 − 1)!
x5!(l − x2 − x5 − 1)!
· l − x2 − x5
l − x2 − x5 − x8
(l − x2 − x5 − 1)!
x8!(l − x2 − x5 − x8 − 1)!
=
l
l − B2
(
l − 1
x2
)(
l − 1 − x2
x5
)(
l − 1 − x2 − x5
x8
)
.
88
Theorem 3.35 With the same hypothesis as in Remark 3.32, we have
Cov(N111/n,N222/n) =
l6
n4(n − 1)2
−
(
1
m3
)2
.
Proof: From Remark 3.32, Remark 3.33 and Remark 3.34 we have
E (N111N222) =








∑
A1+A2+A3=l
∑
x1+x2+x3=A1
∑
x4+x5+x6=A2
∑
x7+x8+x9=A3
x1y5l!
l!
(l − A1)!
l!
(l − A2)!
(n − 2l)!
(n − 2l − A3)!
(
l
x1
)(
l
x2
)(
n − 2l
x3
)(
l − x1
x4
)(
l − x2
x5
)(
n − 2l − x3
x6
)(
l − x1 − x4
x7
)(
l − x2 − x5
x8
)
(
n − 2l − x3 − x6
x9
)]
·









∑
B̂1+B̂2+B̂3=l
∑
y2+y8+y5=B̂2
l!
(l − B1)!
(l − B1 − B̂1)!
(l − B2)!
(l − B2 − B̂2)!
(n − 2l − B3)!
(n − 2l − B3 − B̂3)!
(
l − A2
y5
)(
l − A1
y2
)(
n − 2l − A3
y8
)(
n − l − B̂2
B̂1
)
(
n − l − B̂1 − B̂2
B̂3
)]
·
[
((n − 2l)!)2
(n!)2
]
=








∑
A1+A2+A3=l
∑
x1+x2+x3=A1
∑
x4+x5+x6=A2
∑
x7+x8+x9=A3
l!
l!
(l − A1)!
l!
(l − A2)!
(n − 2l)!
(n − 2l − A3)!
l
(
l − 1
x1 − 1
)(
l
x2
)(
n − 2l
x3
)(
l − x1
x4
)(
l − x2
x5
)(
n − 2l − x3
x6
)(
l − x1 − x4
x7
)(
l − x2 − x5
x8
)
(
n − 2l − x3 − x6
x9
)]
·









∑
B̂1+B̂2+B̂3=l
∑
y2+y8+(y5−1)=B̂2−1
l!
(l − B1)!
(l − B1 − B̂1)!
(l − B2)!
(l − B2 − B̂2)!
(l − A2)
(n − 2l − B3)!
(n − 2l − B3 − B̂3)!
(
l − A2 − 1
y5 − 1
)(
l − A1
y2
)(
n − 2l − A3
y8
)(
n − l − B̂2
B̂1
)
(
n − l − B̂1 − B̂2
B̂3
)]
·
[
((n − 2l)!)2
(n!)2
]
=
(l!)2l(n − l − 1)!
(n − 2l)!
(
n − l − 1
l − 1
)
∑
A1+A2+A3=l
∑
x1+x2+x3=A1
∑
x4+x5+x6=A2
∑
x7+x8+x9=A3
· l!
(l − A1)!
l!
(l − A2)!
(n − 2l)!
(n − 2l − A3)!
(l − A2)(l − B2)
(
l − 1
x1 − 1
)(
l
x2
)(
n − 2l
x3
)
·
(
l − x1
x4
)(
l − x2
x5
)(
n − 2l − x3
x6
)(
l − x1 − x4
x7
)(
l − x2 − x5
x8
)
89
·
(
n − 2l − x3 − x6
x9
) [
((n − 2l)!)2
(n!)2
]
(using Remark 3.33)
=
(l!)2l3(n − l − 1)!
(n − 2l)!
(
n − l − 1
l − 1
)
∑
A1+A2+A3=l
∑
x1+x2+x3=A1
∑
x4+x5+x6=A2
∑
x7+x8+x9=A3
· l!
(l − A1)!
(l − 1)!
(l − A2 − 1)!
(n − 2l)!
(n − 2l − A3)!
(l − A2)(l − B2)
(
l − 1
x1 − 1
)(
l − 1
x2
)(
n − 2l
x3
)
·
(
l − x1
x4
)(
l − 1 − x2
x5
)(
n − 2l − x3
x6
)(
l − x1 − x4
x7
)(
l − 1 − x2 − x5
x8
)
·
(
n − 2l − x3 − x6
x9
) [
((n − 2l)!)2
(n!)2
]
(using Remark 3.34)
=
(l!)2l3(n − l − 1)!
(n − 2l)!
(
n − l − 1
l − 1
)
∑
A1+A2+A3=l
l!
(l − A1)!
(l − 1)!
(l − A2 − 1)!
(n − 2l)!
(n − 2l − A3)!
·
(
n − 2
A1 − 1
)(
n − 1 − A1
A2
)(
n − 1 − A1 − A2
A3
) [
((n − 2l)!)2
(n!)2
]
=
(l!)2l4(n − l − 1)!
(n − 2l)!
(
n − l − 1
l − 1
)
∑
A1+A2+A3=l
(n − 2)!
(n − 1 − l)!
(
l − 1
A1 − 1
)
·
(
l − 1
A2
)(
n − 2l
A3
) [
((n − 2l)!)2
(n!)2
]
=
(l!)2l4(n − l − 1)!
(n − 2l)!
(n − 2)!
(n − 1 − l)!
(
n − 2
l − 1
)(
n − l − 1
l − 1
) [
((n − 2l)!)2
(n!)2
]
=
(l!)2l4(n − l − 1)!(n − 2)!
(n − 2l)!(n − 1 − l)!
(n − 2)!
(l − 1)!(n − l − 1)!
(n − l − 1)!
(l − 1)!(n − 2l)!
((n − 2l)!)2
(n!)2
=
l6
n2(n − 1)2
,
therefore
Cov (N111/n,N222/n) = E ((N111/n)(N222/n)) − E(N111/n)E(N222/n)
=
l6
n4(n − 1)2
−
(
1
m3
)2
.

Remark 3.36 (Case 3) Using the notation in Remark 3.28 and the results in Lemma 3.29, let n
and m be integers greater than or equal to two, where m divides n, and l = n/m. Let N111 be the
random variable indicating the number of observations falling in R111 and let N122 be the random
90
variable indicating the number of observations falling in R122, when we consider a sample of size
n from the product copula, then
P{N111 = x1,N122 = x5} =
∑
A1+A2+A3=l
∑
x2+x3=A1−x1
∑
x4+x6=A2−x5
∑
x7+x8+x9=A3
l!
l!
(l − A1)!
l!
(l − A2)!
(n − 2l)!
(n − 2l − A3)!
(
l
x1
)(
l
x2
)(
n − 2l
x3
)(
l − x1
x4
)(
l − x2
x5
)(
n − 2l − x3
x6
)
(
l − x1 − x4
x7
)(
l − x2 − x5
x8
)(
n − 2l − x3 − x6
x9
)
((n − l)!)2
(n!)2
.
Theorem 3.37 If we consider the same hypothesis of Remark 3.36, then
Cov(N111/n,N122/n) =
l5(l − 1)
n4(n − 1)2
−
(
1
m3
)2
.
Proof: From Remark 3.36, we have
E(N111N122) =
∑
A1+A2+A3=l
∑
x1+x2+x3=A1
∑
x4+x5+x6=A2
∑
x7+x8+x9=A3
l!
l!
(l − A1)!
l!
(l − A2)!
(n − 2l)!
(n − 2l − A3)!
·x1x5
l(l − 1)!
x1!(l − x1)!
l
(l − x2)
(l − 1)!
x2!(l − x2 − 1)!
(
n − 2l
x3
)(
l − x1
x4
)
(l − x2)
· (l − x2 − 1)!
x5!(l − x2 − x5)!
(
n − 2l − x3
x6
)(
l − x1 − x4
x7
)(
l − x2 − x5
x8
)(
n − 2l − x3 − x6
x9
)
· ((n − l)!)2
(n!)2
=
∑
A1+A2+A3=l
∑
x1+x2+x3=A1
∑
x4+x5+x6=A2
∑
x7+x8+x9=A3
l!
l!
(l − A1)!
l!
(l − A2)!
(n − 2l)!
(n − 2l − A3)!
·l2 ((n − l)!)2
(n!)2
[(
l − 1
x1 − 1
)(
l − 1
x2
)(
n − 2l
x3
)] [(
l − x1
x4
)(
l − x2 − 1
x5 − 1
)(
n − 2l − x3
x6
)]
·
[(
l − x1 − x4
x7
)(
l − x2 − x5
x8
)(
n − 2l − x3 − x6
x9
)]
=
∑
A1+A2+A3=l
l!
l!
(l − A1)!
l!
(l − A2)!
(n − 2l)!
(n − 2l − A3)!
l2 ((n − l)!)2
(n!)2
(
n − 2
A1 − 1
)
·
(
n − 1 − A1
A2 − 1
)(
n − A1 − A2
A3
)
91
=
l!l2((n − l)!)2
(n!)2
∑
A1+A2+A3=l
(
l!
(l − A1)!
l!
(l − A2)!
(n − 2l)!
(n − 2l − A3)!
(n − 2)!
(A1 − 1)!(n − 1 − A1)!
(n − 1 − A1)!
(A2 − 1)!(n − A1 − A2)!
(n − A1 − A2)!
A3!(n − l)!
)
=
l!l2((n − l)!)2
(n!)2
l2(n − 2)!
(n − l)!
∑
A1+A2+A3=l
(
(l − 1)!
(A1 − 1)!(l − A1)!
(l − 1)!
(A2 − 1)!(l − A2)!
(n − 2l)!
A3!(n − 2l − A3)!
)
=
l!l4((n − l)!)2(n − 2)!
(n!)2(n − l)!
∑
A1+A2+A3=l
(
l − 1
A1 − 1
)(
l − 1
A2 − 1
)(
n − 2l
A3
)
=
l!l4((n − l)!)2(n − 2)!
(n!)2(n − l)!
(
n − 2
l − 2
)
=
l!l4((n − l)!)2(n − 2)!
(n!)2(n − l)!
(n − 2)!
(l − 2)!(n − l)!
=
l5(l − 1)
n2(n − 1)2
,
therefore
Cov(N111/n,N122/n) =
l5(l − 1)
n4(n − 1)2
−
(
1
m3
)2
.

The following theorem describes the joint probability distribution of the boxes generated for the
uniform partition of size m, with m ≥ 2, when m divides n, in the three dimensional case.
Theorem 3.38 Let m ≥ 2, n ∈ N, where m divides n, l = n/m and Im = {1, · · ·m}, let Ri jk,
with i, j, k ∈ Im, be the uniform partition of size m of I3 = [0, 1]3 and Ni jk the random variables
that indicates the number of observations falling in Ri jk, respectively, for all i, j, k ∈ Im, when we
consider a sample of size n from the product copula; let ni jk, with i, j, k ∈ Im, be zero or positive
integer satisfying the following restrictions
m
∑
j,k=1
ni jk = l (for all i ∈ Im),
m
∑
i,k=1
ni jk = l (for all j ∈ Im),
m
∑
i, j=1
ni jk = l (for all k ∈ Im),
92
then
P









⋂
i, j,k∈Im
{Ni jk = ni jk}









=
(l!)3m
(n!)2
∏
i, j,k∈Im
ni jk!
.
Proof: Similarly to the proof of the Theorem 3.17, we will use the counting methodology provided
on the Remark 3.3, using permutations for the counting of the third coordinate Z. The order count
is
R111,R112, · · · ,R11m, · · · ,R1m1,R1m2, · · · ,R1mm
R211,R212, · · · ,R21m, · · · ,R2m1,R2m2, · · · ,R2mm
...
Rm11,Rm12, · · · ,Rm1m, · · · ,Rmm1,Rmm2, · · · ,Rmmm.
The number of possibilities for l observations in the region R̂1 = [0, l/n]× [0, 1]× [0, 1] is given by
N̂1 =
[(
l
n111
)
Pl
n111
Pl
n111
(
l − n111
n112
)
Pl−n111
n112
Pl
n112
· · ·
(
l −∑m−1
k=1 n11k
n11m
)
P
l−∑m−1
k=1 n11k
n11m
Pl
n11m
]
[(
l −∑m
k=1 n11k
n121
)
Pl
n121
Pl−n111
n121
(
l − n121 −
∑m
k=1 n11k
n122
)
Pl−n121
n122
Pl−n112
n122
· · ·
(
l −∑m−1
k=1 n12k −
∑m
k=1 n11k
n12m
)
P
l−∑m−1
k=1 n12k
n12m
Pl−n11m
n12m
]
· · ·
[(
l −∑m−1
j=1
∑m
k=1 n1 jk
n1m1
)
Pl
n1m1
P
l−∑m−1
j=1 n1 j1
n1m1
(
l − n1m1 −
∑m−1
j=1
∑m
k=1 n1 jk
n1m2
)
Pl−n1m1
n1m2
P
l−∑m−1
j=1 n1 j2
n1m2
· · ·
(
l −∑m−1
k=1 n1mk −
∑m−1
j=1
∑m
k=1 n1 jk
n1mm
)
P
l−∑m−1
k=1 n1mk
n1mm
P
l−∑m−1
j=1 n1 jk
n1mm
]
,
and the number of possibilities for l observations in the region R̂s = (l(s−1)/n, ls/n]×[0, 1]×[0, 1],
for all s ∈ 2, · · · ,m, is given by
N̂s =
[(
l
ns11
)
P
l−∑s−1
i=1
∑m
k=1 ni1k
ns11
P
l−∑s−1
i=1
∑m
j=1 ni j1
ns11
(
l − ns11
ns12
)
P
l−ns11−
∑s−1
i=1
∑m
k=1 ni1k
ns12
P
l−∑s−1
i=1
∑m
j=1 ni j2
ns12
· · ·
(
l −∑m−1
k=1 ns1k
ns1m
)
P
l−∑m−1
k=1 ns1k−
∑s−1
i=1
∑m
k=1 ni1k
ns1m
P
l−∑s−1
i=1
∑m
j=1 ni jm
ns1m
] [(
l −∑m
k=1 ns1k
ns21
)
P
l−∑s−1
i=1
∑m
k=1 ni2k
ns21
P
l−ns11−
∑s−1
i=1
∑m
j=1 ni j1
ns21
(
l − ns21 −
∑m
k=1 ns1k
ns22
)
P
l−ns21−
∑s−1
i=1
∑m
k=1 ni2k
ns22
P
l−ns12−
∑s−1
i=1
∑m
j=1 ni j2
ns22
· · ·
93
(
l −∑m
k=1 ns1k −
∑m−1
k=1 ns2k
ns2m
)
P
l−∑m−1
k=1 ns2k−
∑s−1
i=1
∑m
k=1 ni2k
ns2m
P
l−ns1m−
∑s−1
i=1
∑m
j=1 ni jm
ns2m
]
· · ·
[(
l −∑m
j,k=1 ns jk
nsm1
)
P
l−∑s−1
i=1
∑m
k=1 nimk
nsm1
P
l−∑m−1
j=1 ns j1−
∑s−1
i=1
∑m
j=1 ni j1
nsm1
(
l − nsm1 −
∑m
j,k=1 ns jk
nsm2
)
P
l−nsm1−
∑s−1
i=1
∑m
k=1 nimk
nsm2
P
l−∑m−1
j=1 ns j2−
∑s−1
i=1
∑m
j=1 ni j2
nsm2
· · ·
(
l −∑m−1
k=1 nsm j −
∑m
j,k=1 ns jk
nsmm
)
P
l−∑m−1
k=1 nsmk−
∑s−1
i=1
∑m
k=1 nimk
nsmm
P
l−∑m−1
j=1 ns jm−
∑s−1
i=1
∑m
j=1 ni jm
nsmm
]
we have that
N̂1 ·
m
∏
s=2
N̂s =
(l!)3m
∏
i, j,k∈Im
ni jk!
.
We divided between (n!)2, the total number of possibilities in which we can see n points in I3 =
[0, 1]3, corresponding to the modified sample of the a sample of size n from the product copula,
therefore
P









⋂
i, j,k∈Im
{Ni jk = ni jk}









=
(l!)3m
(n!)2
∏
i, j,k∈Im
ni jk!
.

3.3 Summary of results and generalizations
We present a summary of the precedent results and the generalizations of the previous expressions
for the moments in dimension greater than three.
1. Dimension two. Let m ≥ 2, n ∈ N, where m divides n, and l = n/m, let N11, N12 and N22 be
the random variables that indicates the number of observations on the boxes R11 = [0, l/n]2,
R12 = [0, l/n]×(l/n, 2l/n] and R22 = (l/n, 2l/n]2, respectively, when we consider the modified
sample of a sample of size n from the product copula.
(a) First Moment.
E(N11/n) =
1
m2
(b) Second Moment.
E((N11/n)2) =
l2(l − 1)2
n3(n − 1)
+
1
nm2
94
(c) Variance.
Var(N11/n) =
l2(l − 1)2
n3(n − 1)
+
1
nm2
− 1
m4
(d) Covariance I.
Cov(N11/n,N12/n) =
l3(l − 1)
n3(n − 1)
− 1
m4
(e) Covariance II.
Cov(N11/n,N22/n) =
l4
n3(n − 1)
− 1
m4
2. Dimension three. Let m ≥ 2, n ∈ N, where m divides n, and l = n/m, let N111, N112, N122
and N222 be the random variables that indicates the number of observations on the boxes
R111 = [0, l/n]3, R112 = [0, l/n]× [0, l/n]× (l/n, 2l/n], R122 = [0, l/n]× (l/n, 2l/n]× (l/n, 2l/n]
and R222 = (l/n, 2l/n]3, respectively, when we consider the modified sample of a sample of
size n from the product copula.
(a) First Moment.
E(N111/n) =
1
m3
(b) Second Moment.
E((N111/n)2) =
l3(l − 1)3
n4(n − 1)2
+
1
nm3
(c) Variance.
Var(N111/n) =
l3(l − 1)3
n4(n − 1)2
+
1
nm3
− 1
m6
(d) Covariance I.
Cov(N111/n,N112/n) =
l4(l − 1)2
n4(n − 1)2
− 1
m6
(e) Covariance II.
Cov(N111/n,N122/n) =
l5(l − 1)
n4(n − 1)2
− 1
m6
(f) Covariance III.
Cov(N111/n,N222/n) =
l6
n4(n − 1)2
− 1
m6
95
3. General case. Let m ≥ 2, n ∈ N, where m divides n, l = n/m, d ≥ 2, 1 = (1, · · · , 1) (d
times) and 2 = (2, · · · , 2) (d times), let N1 be the random variable that indicates the number
of observations on the box R1 = [0, l/n]d, when we consider the modified sample of a sample
of size n from the product copula.
(a) First Moment.
E(N1/n) =
1
md
(b) Second Moment.
E((N1/n)2) =
ld(l − 1)d
nd+1(n − 1)d−1
+
1
nmd
(c) Variance.
Var(N1/n) =
ld(l − 1)d
nd+1(n − 1)d−1
+
1
nmd
− 1
m2d
(d) Covariance I.
Cov(N1/n,N2/n) =
l2d
nd+1(n − 1)d−1
− 1
m2d
where N2 is the random variable that indicates the number of observations on the box R2 =
(l/n, 2l/n]d.
(e) Covariance II. Let j = ( j1, · · · , jd) ∈ {1, 2}d, j , 1, 2, if we define
R j = Î1 × · · · × Îd
where
Îi =
{
[0, l/n] i f ji = 1
(l/n, 2l/n] i f ji = 2
for i ∈ {1, · · · , d}, then
Cov(N1/n,N j/n) =
l2d−k(l − 1)k
nd+1(n − 1)d−1
− 1
m2d
.
where N j is the random variable that indicates the number of observations on the box R j and
k the number of coordinates equals to one in j.
(f) Joint distribution. Let d ≥ 2, and let Ni1···id be the random variable that indicates the
number of observations in the box Ri1···id = 〈(i1 − 1)/m, i1/m]×· · ·×〈(id − 1)/m, id/m], where
96
i1, · · · , id ∈ Im and the notation “〈” indicates “(” if ik − 1 > 0 and “[” if ik − 1 = 0, for all
k ∈ Id, when we consider the modified sample of size n from the product copula Πd. Then
P







⋂
i1,···,id∈Im
{Ni1···id = ni1···id}







=
(l!)dm
(n!)d−1
∏
i1,···,id∈Im
ni1...id !
.
3.4 Case: m does not divide n
In the next pages we show that we can obtain similar results to what was done previously, now
without the hypothesis m divides n. We exemplify with the two dimensional case.
The following results establish the properties of random variables associated with the observations
in the boxes from a not uniform partition of order m, that is, a partition of the box I2 = [0, 1]2
generated by dividing the two intervals I = [0, 1] in the Cartesian product into m parts, where m
does not divide n.
Definition 3.39 Let n ≥ 2, 0 < l1 < l2 < n and 0 < j1 < j2 ≤ n be integers with j2 − j1 = l2. We
define the following regions in the unit square I2 = [0, 1]2
R1 j = [0, l1/n] × ( j1/n, j2/n
]
R1 = ([0, l1/n] × [0, 1])\R1 j
R = (l1/n, 1] × [0, 1] .
Remark 3.40 In the following results we consider than in the unit square I2 = [0, 1]2 there exists
n points corresponding to the range statistics from a sample of size n from the product copula.
Lemma 3.41 Let k1 be the number of points in R1 j and x1 the number of points in R1, we have that
∑
k1+x1=l1
(
l1
k1
)
P
l2
k1
(
l1 − k1
x1
)
Pn−l2
x1
= Pn
l1
.
Proof: We can observe that
(
l1 − k1
x1
)
=
(
x1
x1
)
= 1
then
∑
k1+x1=l1
(
l1
k1
)
P
l2
k1
(
l1 − k1
x1
)
Pn−l2
x1
=
∑
k1+x1=l1
(
l1
k1
)
P
l2
k1
Pn−l2
x1
97
=
∑
k1+x1=l1
l1!
k1!(l1 − k1)!
l2!
(l2 − k1)!
(n − l2)!
(n − l2 − x1)!
=
∑
k1+x1=l1
l1!
k1!x1!
l2!
(l2 − k1)!
(n − l2)!
(n − l2 − x1)!
= l1!
∑
k1+x1=l1
(
l2
k1
)(
n − l2
x1
)
= l1!
(
n
l1
)
(Vandermonde’s identity)
= Pn
l1
.

Lemma 3.42 Let N1 j be the random variable that indicates the number of observations falling in
R1 j, then the following equality holds
P{N1 j = k1} =
l1!(n − l1)!
n!
(
l2
k1
)(
n − l2
l1 − k1
)
=
(
l2
k1
)(
n−l2
l1−k1
)
(
n−l1
l1
)
this probability corresponds to a hypergeometric distribution with parameters: n the population
size, l1 the class size and l2 the sample size.
Proof: We obtained the equality considering
∑
k1+x1=l1
(
l1
k1
)
P
l2
k1
(
l1 − k1
x1
)
Pn−l2
x1
= l1!
∑
k1+x1=l1
(
l2
k1
)(
n − l2
x1
)
from the proof of Lemma 3.42, this equality indicates the number of ways in which we can have
k1+ x1 points in the region R1 j∪R1; setting the value k1, this result is multiplied by (n− l1)! (number
of ways in which we can have n− l1 points in the region R discarding l1 possibilities corresponding
to the coordinates occupied by the observations in the region R1 j ∪ R1) and divided by n!, the total
number of ways that we can observe n points in I2 = [0, 1]2. 
Theorem 3.43 Let N1 j be the random variable that indicates the number of observations falling
in R1 j when we consider a sample of size n from the product copula, then
E(N1 j/n) =
l1l2
n2
, E((N1 j/n)2) =
l2(l2 − 1)l1(l1 − 1)
n3(n − 1)
+
l1l2
n3
98
and
Var(N1 j/n) =
l2(l2 − 1)l1(l1 − 1)
n3(n − 1)
+
l1l2
n3
−
(
l1l2
n2
)2
.
Proof:
E(N1 j) =
l1
∑
k1=0
k1
l1!(n − l1)!
n!
(
l2
k1
)(
n − l2
l1 − k1
)
=
l1!l2(n − l1)!
n!
l1
∑
k1=1
(
l2 − 1
k1 − 1
)(
n − l2
l1 − k1
)
=
l1!l2(n − l1)!
n!
l1−1
∑
u1=0
(
l2 − 1
u1
)(
n − l2
l1 − 1 − u1
)
(u1 = k1 − 1)
=
l1!l2(n − l1)!
n!
(
n − 1
l1 − 1
)
=
l1!l2(n − l1)!
n!
(n − 1)!
(l1 − 1)!(n − l1)!
=
l1l2
n
and
E(N1 j/n) =
l1l2
n2
.
We have,
E(N2
1 j) =
l1
∑
k1=0
k2
1
l1!(n − l1)!
n!
(
l2
k1
)(
n − l2
l1 − k1
)
=
l1
∑
k1=0
(k1(k1 − 1) + k1)
l1!(n − l1)!
n!
(
l2
k1
)(
n − l2
l1 − k1
)
=
l1
∑
k1=2
k1(k1 − 1)
l1!(n − l1)!
n!
(
l2
k1
)(
n − l2
l1 − k1
)
+
l1
∑
k1=0
k1
l1!(n − l1)!
n!
(
l2
k1
)(
n − l2
l1 − k1
)
=
l1
∑
k1=2
l1!(n − l1)!
n!
l2(l2 − 1)
(
l2 − 2
k1 − 2
)(
n − l2
l1 − k1
)
+
l1l2
n
=
l1!(n − l1)!
n!
l2(l2 − 1)
l1−2
∑
u1=0
(
l2 − 2
u1
)(
n − l2
(l1 − 2) − u1
)
+
l1l2
n
(u1 = k1 − 2)
99
=
l1!(n − l1)!
n!
l2(l2 − 1)
(
n − 2
l1 − 2
)
+
l1l2
n
=
l2(l2 − 1)l1(l1 − 1)
n(n − 1)
+
l1l2
n
.
Therefore
E((N1 j/n)2) =
l2(l2 − 1)l1(l1 − 1)
n3(n − 1)
+
l1l2
n3
and
Var(Ni j/n) = E((Ni j/n)2) − (E(Ni j/n))2
=
l2(l2 − 1)l1(l1 − 1)
n3(n − 1)
+
l1l2
n3
−
(
l1l2
n2
)2
.

In a similar way to the case where m divides n, the calculus of the covariances is divided in two
cases. The following result corresponds to the first one.
Definition 3.44 Let n ≥ 2, 0 < l1, l2, l3 < n and 0 < j1 < j2 < j3 ≤ n be integers with j2 − j1 = l2
and j3 − j2 = l3. We define the following regions in the unit square I2 = [0, 1]2
R1 j1 = [0, l1/n] × ( j1/n, j2/n
]
R1 j2 = [0, l1/n] × ( j2/n, j3/n
]
R1 = ([0, l1/n] × [0, 1])\(R1 j1 ∪ R1 j2)
R = (l1/n, 1] × [0, 1] .
Lemma 3.45 Let k1 be the number of points in R1 j1 , k2 the number of points in R1 j2 and x1 the
number of points in R1, then
∑
k1+k2+x1=l1
(
l1
k1
)
P
l2
k1
(
l1 − k1
k2
)
P
l3
k2
(
l1 − k1 − k2
x1
)
Pn−l2−l3
x1
= Pn
l1
.
Proof: We have that
∑
k1+k2+x1=l1
(
l1
k1
)
P
l2
k1
(
l1 − k1
k2
)
P
l3
k2
(
l1 − k1 − k2
x1
)
Pn−l2−l3
x1
=
∑
k1+k2+x1=l1
l1!
k1!k2!x1!
P
l2
k1
P
l3
k2
Pn−l2−l3
x1
=
∑
k1+k2+x1=l1
l1!
(
l2
k1
)(
l3
k2
)(
n − l2 − l3
x1
)
100
= l1!
(
n
l1
)
= Pn
l1
.

Remark 3.46 Let N1 j1 be the random variable that indicates the number of observations falling in
R1 j1 and let N1 j2 be the random variable that indicates the number of observations falling in R1 j2
then
P{N1 j1 = k1,N1 j2 = k2} =
l1!(n − l1)!
n!
(
l2
k1
)(
l3
k2
)(
n − l2 − l3
l1 − k1 − k2
)
.
Lemma 3.47 Whit the same hypothesis of Remark 3.46, we have
Cov(Ni j1/n,Ni j2/n) =
l1(l1 − 1)l2l3
n2(n − 1)
−
(
l1l2
n2
) (
l1l3
n2
)
.
Proof:
E(N1 j1 N1 j2) =
l1
∑
k1+k2=0
k1k2
l1!(n − l1)!
n!
(
l2
k1
)(
l3
k2
)(
n − l2 − l3
l1 − k1 − k2
)
=
l1
∑
k1+k2=0
l1!(n − l1)!
n!
l2l3
(
l2 − 1
k1 − 1
)(
l3 − 1
k2 − 1
)(
n − l2 − l3
l1 − k1 − k2
)
=
l1!(n − l1)!l2l3
n!
l1
∑
k1+k2=0
(
l2 − 1
k1 − 1
)(
l3 − 1
k2 − 1
)(
n − l2 − l3
l1 − k1 − k2
)
=
l1!(n − l1)!l2l3
n!
(
n − 2
l1 − 2
)
=
l1(l1 − 1)l2l3
n(n − 1)
,
and
E((N1 j1/n)(N1 j2/n)) =
l1(l1 − 1)l2l3
n3(n − 1)
.
Therefore
Cov(N1 j1/n,N1 j2/n) = E((N1 j1/n)(N1 j2/n)) − E(N1 j1/n)E(N1 j2/n)
=
l1(l1 − 1)l2l3
n3(n − 1)
−
(
l1l2
n2
) (
l1l3
n2
)
.
101

We use the next results for the calculation of the covariances for the second case.
Definition 3.48 Let n ≥ 2, 0 < l1, l2, l3, l4 < m, 0 < j1 < j2 < j3 ≤ n and 0 = i1 < i2 < i3 ≤ n be
integers with i2 − i1 = l1, i3 − i2 = l3, j2 − j1 = l2 and j3 − j2 = l4. We define the following regions
in the unit square I2 = [0, 1]2
R1 j1 = [0, l1/n] × ( j1/n, j2/n
]
R1 j2 = [0, l1/n] × ( j2/n, j3/n
]
R2 j1 = (i2/n, i3/n] × ( j1/n, j2/n
]
R2 j2 = (i2/n, i3/n] × ( j2/n, j3/n
]
R1 = ([0, l1/n] × [0, 1])\(R1 j1 ∪ R1 j2)
R2 = ((i2/n, i3/n] × [0, 1])\(R2 j1 ∪ R2 j2).
Lemma 3.49 Let k1 be the number of points in R1 j1 , x1 the number of points in R1 j2 , x2 the number
of points in R1, y1 the number of points in R2 j1 , k2 the number of points in R2 j2 and y2 the number
of points in R2, then
∑
k1+x1+x2=l1
∑
y1+k2+y2=l3
(
l1
k1
)(
l1 − k1
x1
)(
l1 − k1 − x1
x2
)
P
l2
k1
Pl4
x1
Pn−l2−l4
x2
·
(
l3
k2
)(
l3 − k2
y1
)(
l3 − k2 − y1
y2
)
Pl2−k1
y1
P
l4−x1
k2
Pn−l2−l4−x2
y2
= Pn
l1
P
n−l1
l3
.
Proof:
∑
k1+x1+x2=l1
∑
y1+k2+y2=l3
(
l1
k1
)(
l1 − k1
x1
)(
l1 − k1 − x1
x2
)
P
l2
k1
Pl4
x1
Pn−l2−l4
x2
·
(
l3
k2
)(
l3 − k2
y1
)(
l3 − k2 − y1
y2
)
Pl2−k1
y1
P
l4−x1
k2
Pn−l2−l4−x2
y2
=
∑
k1+x1+x2=l1
∑
y1+k2+y2=l3
(
l1!
k1!x1!x2!
P
l2
k1
Pl4
x1
Pn−l2−l4
x2
) (
l3!
k2!y1!y2!
Pl2−k1
y1
P
l4−x1
k2
Pn−l2−l4−x2
y2
)
=
∑
k1+x1+x2=l1
∑
y1+k2+y2=l3
l1!l3!
(
l2
k1
)(
l4
x1
)(
n − l2 − l4
x2
)(
l2 − k1
y1
)(
l4 − x1
k2
)(
n − l2 − l4 − x2
y2
)
=
∑
k1+x1+x2=l1
l1!l3!
(
l2
k1
)(
l4
x1
)(
n − l2 − l4
x2
)(
n − l1
l3
)
102
= l1!l3!
(
n
l1
)(
n − l1
l3
)
= Pn
l1
P
n−l1
l3
.

Remark 3.50 Let N11 j1 be the random variable indicating the number of observations falling in
R11 j1 and Ni2 j2 , the random variable indicating the number of observations falling in Ri2 j2 , then
P{Ni1 j1 = k1,Ni2 j2 = k2} =
(n − l1 − l3)!
n!
∑
x1+x2=l1−k1
∑
y1+y2=l3−k2
l1!l3!
(
l2
k1
)(
l4
x1
)(
n − l2 − l4
x2
)
·
(
l2 − k1
y1
)(
l4 − x1
k2
)(
n − l2 − l4 − x2
y2
)
.
Lemma 3.51 With the same hypothesis of Remark 3.50, we have
Cov(Ni1 j1/n,Ni2 j2/n) =
l1l2l3l4
n3(n − 1)
−
(
l1l2
n2
) (
l3l4
n2
)
.
Proof:
E(Ni1 j1 Ni2 j2) =
(n − l1 − l3)!
n!
∑
k1+x1+x2=l1
∑
y1+k2+y2=l3
k1k2l1!l3!
(
l2
k1
)(
l4
x1
)(
n − l2 − l4
x2
)
·
(
l2 − k1
y1
)(
l4 − x1
k2
)(
n − l2 − l4 − x2
y2
)
=
(n − l1 − l3)!
n!
∑
k1+x1+x2=l1
∑
y1+k2+y2=l3
l1!l3!l2
(
l2 − 1
k1 − 1
)
l4
l4 − x1
(
l4 − 1
x1
)(
n − l2 − l4
x2
)
·
(
l2 − k1
y1
)
(l4 − x1)
(
l4 − x1 − 1
k2 − 1
)(
n − l2 − l4 − x2
y2
)
=
(n − l1 − l3)!
n!
∑
k1+x1+x2=l1
∑
y1+k2+y2=l3
l1!l3!l2l4
(
l2 − 1
k1 − 1
)(
l4 − 1
x1
)(
n − l2 − l4
x2
)
·
(
l2 − k1
y1
)(
l4 − x1 − 1
k2 − 1
)(
n − l2 − l4 − x2
y2
)
=
(n − l1 − l3)!
n!
∑
k1+x1+x2=l1
l1!l3!l2l4
(
l2 − 1
k1 − 1
)(
l4 − 1
x1
)(
n − l2 − l4
x2
)(
n − 1 − l1
l3 − 1
)
=
(n − l1 − l3)!
n!
l1!l3!l2l4
(
n − 2
l1 − 1
)(
n − 1 − l1
l3 − 1
)
103
=
(n − l1 − l3)!
n!
l1!l3!l2l4
(n − 2)!
(l1 − 1)!(n − l1 − 1)!
(n − 1 − l1)!
(l3 − 1)!(n − l1 − l3)!
=
l1l2l3l4
n(n − 1)
,
and
E((Ni1 j1/n)(Ni2 j2/n)) =
l1l2l3l4
n3(n − 1)
.
Finally
Cov(Ni1 j1/n,Ni2 j2/n) = E((Ni1 j1/n)(Ni2 j2/n)) − E(Ni1 j1/n)E(Ni2 j2/n)
=
l1l2l3l4
n3(n − 1)
−
(
l1l2
n2
) (
l3l4
n2
)
.

Finally, the next theorem describes the joint probability distribution of the boxes generated by the
not uniform partition of size m, with m ≥ 2, of I2 = [0, 1]2.
Theorem 3.52 Let m ≥ 2, n ∈ N, where m not necessarily divides n, and Ik = {1, · · · k}, k ∈ N, we
consider {t0, t1, · · · , tm} ⊂ In with 0 = t0 < t1 < · · · < tm = n. We define li = ti − ti−1, for all i ∈ Im
and the partition, not necessarily uniform, of size m of I2 = [0, 1]2 by
Ri j =
〈
ti−1
n
,
ti
n
]
×
〈 t j−1
n
,
t j
n
]
for all i, j ∈ Im, the notation “〈” indicates “(” if tk−1 > 0 and “[” if tk−1 = 0, for all k ∈ Im.
Let Ni j be the random variables that indicates the number of observations falling in Ri j, respec-
tively, for all i, j ∈ Im, when we consider a sample of size n from the product copula; let ni j, with
i, j ∈ Im, be zero or a positive integer that satisfies the following restrictions
m
∑
j=1
ni j = li (for all i ∈ Im),
m
∑
i=1
ni j = l j (for all j ∈ Im),
and
m
∑
k=1
lk = n
then
P







⋂
i, j∈Im
{Ni j = ni j}







=
∏m
i=1(li!)
2
n!
∏
i, j∈Im
ni j!
.
104
Proof: We use the counting methodology provided on the Remark 3.3 for the regions Ri j, i, j ∈ Im,
in the following order
R11, · · · ,R1m,R21, · · · ,R2m,Rm1, · · · ,Rmm.
The number of ways in which we can observe ni j points in each region Ri j, i, j ∈ Im, is given by
(
l1
n11
)
Pl1
n11
(
l1 − n11
n12
)
Pl2
n12
· · ·
(
l1 −
∑m−1
j=1 n1 j
n1m
)
Plm
n1m
·
(
l2
n21
)
Pl1−n11
n21
(
l2 − n21
n22
)
Pl2−n12
n22
· · ·
(
l2 −
∑m−1
j=1 n2 j
n2m
)
Plm−n1m
n2m
...
·
(
lm
nm1
)
P
l1−
∑m−1
i=1 ni1
nm1
(
lm − nm1
nm2
)
P
l2−
∑m−1
i=1 ni2
nm2
· · ·
(
lm −
∑m−1
j=1 nm j
nmm
)
P
lm−
∑m−1
i=1 nim
nmm
=
∏m
i=1(li!)
2
∏
i, j∈Im
ni j!
,
and we divided between n!, the total number of possibilities in which we can see n points in
I2 = [0, 1], corresponding to the modified sample of the original sample of size n from the product
copula, therefore
P







⋂
i, j∈Im
{Ni j = ni j}







=
∏m
i=1(li!)
2
n!
∏
i, j∈Im
ni j!
.

105
4 Convergence results of the sample copula
In this last chapter, we study the weak convergence of the sample copula process
√
n(Cn
m − C), in
this way we will have, for the sample copula, a weak convergence theorem similar to that for the
empirical copula. This chapter is divided into three parts: In the first part we give a review of the
results in the theory of empirical processes, and the results of the weak convergence to a Gaussian
process of the empirical process
√
n(Cn − C) given in [13] because, using this convergence, we
obtain the corresponding convergence of the sample copula process; in the second part, and in
order to be able to apply the results given in the first part, we show that the sample copula can
be represented as a linear functional evaluated in the empirical copula, under this representation
we have Hadamard’s differentiation and then we can apply the delta method. In the third part we
perform several simulations of the sample copula process at a given point to analyze the properties
of the convergence to the Gaussian process with a given variance-covariance structure.
4.1 Weak convergence of empirical process
The weak convergence of an empirical process is studied extensively in Billingsley’s book [2] in
the case of processes with parameter space I = [0, 1]. The extensions of these results, that is,
processes with parameter space equal to Id = [0, 1]d, d ≥ 1, can be found in [38]. We begin this
section with some of the main definitions and results of [38].
Definition 4.1 We define the unit cube in Rd by
Ed = [0, 1] × · · · × [0, 1] (d times)
for t ∈ Ed, with t = (t1, · · · , td), it is considered
|t| = max{|ti| | i = 1, · · · , d}.
We define
P = {ρ = (ρi, · · · , ρd) | ρi = 0 ó ρi = 1, i = 1, · · · , d}
as the set of the 2d vertices of Ed.
Definition 4.2 Let t ∈ Ed and ρ ∈ P, we define the quadrants Q(ρ, t) and Q̂(ρ, t) in Ed with vertex
t as follows
Q(ρ, t) = I(ρ1, t1) × · · · × I(ρd, td)
where
I(r, s) =
{
[0, s) si r = 0
(s, 1] si r = 1
106
and
Q̂(ρ, t) = Î(ρ1, t1) × · · · × Î(ρd, td)
where
Î(r, s) =















[0, s) si r = 0 y s < 1
[0, 1] si r = 0 y s = 1
∅ si r = 1 y s = 1
[s, 1] si r = 1 y s < 1.
Example 4.3 To exemplify the Definition 4.2, we consider d = 3, then
E3 = [0, 1] × [0, 1] × [0, 1],
let t ∈ E3, with t = (0.5, 0.8, 0) and ρ ∈ P, with ρ = (0, 1, 1). We have
Q(ρ, t) = I(0, 0.5) × I(1, 0.8) × I(1, 0)
with
I(0, 0.5) = [0, 0.5)
I(1, 0.8) = (0.8, 1]
I(1, 0) = (0, 1]
therefore,
Q(ρ, t) = [0, 0.5) × (0.8, 1] × (0, 1].
On the other hand
Q̂(ρ, t) = Î(0, 0.5) × Î(1, 0.8) × Î(1, 0)
with
Î(0, 0.5) = [0, 0.5)
Î(1, 0.8) = [0.8, 1]
Î(1, 0) = [0, 1]
therefore,
Q̂(ρ, t) = [0, 0.5) × [0.8, 1] × [0, 1].
We note that Q̂ is the closure, on the left side, of the intervals that define Q.
Remark 4.4 The quadrants Q(ρ, t) and Q̂(ρ, t), called quadrants of continuity, satisfy the follow-
ing properties
107
1. Q(ρ, t) ⊂ Q̂(ρ, t) ⊂ Q(ρ, t) (the notation A indicates the closure of set A).
2. Q(ρ, t) = ∅ if and only if Q̂(ρ, t) = ∅.
3. Q̂(ρ, t) ∩ Q̂(ρ′, t) = ∅ if ρ , ρ′.
4.
⋃
ρ∈P Q̂(ρ, t) = Ed (t ∈ Ed),
5. For each t ∈ Ed exists only a vertex ρ = ρ(t), denoted by σ, such as t ∈ Q̂(σ, t).
6. int(Q(σ, t)) , ∅ y Q̂(σ, t) = Q(σ, t).
Definition 4.5 Let f : Ed −→ R be a function. If for t ∈ Ed, ρ ∈ P, with Q(ρ, t) , ∅, and for every
sequence {tn}n≥1 ⊂ Q(ρ, t), with tn −→ t, the sequence { f (tn)}n≥1 converges, then the only limit is
denoted by f (t + 0ρ) and is called ρ-limit or quadrant limit.
Definition 4.6 The D[0, 1]d space is define as the set of all functions f : Ed −→ R for which the
ρ-limit of f in t, with t ∈ Ed, exists for all ρ ∈ P such that Q(ρ, t) , ∅ and they are continuous in
the following sense: f (t) = f (t + 0σ).
Definition 4.7 We denote by Λ the set of all functions
λ : [0, 1] −→ [0, 1]
increasing, continuous and surjectives. For λ̂ = (λ1, · · · , λd) ∈ Λd = Λ × · · · × Λ (d times) and
t = (t1, · · · , td) ∈ Ed we define λ̂(t) = (λ1(t1), · · · , λd(td)).
Definition 4.8 For µ ∈ Λ we define
‖µ‖ = sup
u,v∈[0,1],u,v
{
∣
∣
∣
∣
∣
∣
log
(
µ(u) − µ(v)
u − v
)
∣
∣
∣
∣
∣
∣
}
and we define the metric d0 in D[0, 1]d by
d0( f , g) = inf
{
ε > 0 | there exists λ̂ ∈ Λd with ‖λi‖ ≤ ε, i ∈ Id, and sup
t∈Ed
| f (t) − g(λ(t))| ≤ ε
}
for f , g ∈ D[0, 1]d.
Remark 4.9 The metric space (D[0, 1]d, d0) is a Polish space.
108
Since the empirical distributions functions are functions that have discontinuities, then these are
elements of the metric space (D[0, 1]d, d0). We have that the space C[0, 1]d, that is the space of
the real continuous functions with domain Id = [0, 1]d, is a subset of D[0, 1]d, and in this case,
from [2] and [38], we can establish the equivalence between the d0 metric, given in Definition 4.8
and the supreme metric given by dsup( f , g) =
∑
t∈[0,1]d | f (t) − g(t)| for all f , g ∈ C[0, 1]d. Among
the advantages of using the sample copula rather than empirical copula, is that the sample copula
is a continuous function, that is, for d ≥ 2, the d-sample copula is an element of the metric
spaces C[0, 1]d and D[0, 1]d, while the empirical copula in dimension d, being a function with
discontinuities of the first order, is only an element of the space D[0, 1]d.
The next results given in [38], are the basis for the study of weak convergence of the empirical
copula process under the independence case.
Remark 4.10 We consider U j = (U j1, · · · ,U jd), j ∈ N, random vector independent and identically
distributed, where U ji is an uniform random variable in (0, 1), j ∈ N and i ∈ {1, · · · , d}. For each
j ∈ N the distribution function F of U j satisfies the Lipschitz’s condition
|F(t) − F(t′)| ≤ K|t − t′|
with a constant K and t, t′ ∈ [0, 1]d.
Definition 4.11 Let n ∈ N, we define the random variables Yn in D[0, 1]d by
Yn(t) =
1
n
1
2








n
∑
j=1
d
∏
i=1
(1[0,ti](U ji) − ti)








.
Theorem 4.12 Let r ≥ 1 and let t1, · · · , tr ∈ Ed, under the assumption of independence in the
components of the random vector U j, j ∈ N, we have
(Yn(t1), . . . ,Yn(tr))
D−→ N(0̂, γY(t1, · · · , tr))
with 0̂ the zero vector in Ed and
γY(t1, · · · , tr) =







d
∏
i=1
((tνi ∧ tµi) − (tνitµi))







ν,µ=1,···,r
The next definitions and results provide the principles to formulate the convergence of the empirical
processes using Hadamard’s derivative and the delta method, these methodologies allows us to
109
study the convergence of an empirical process using a Taylor’s first order approximation applied
to a linear functional and the convergence in probability of the residual of the approximation from
the Slustky’s theorem. The delta method in empirical process with one dimension of the parameter
space is shown in [47] and the generalization for empirical process with higher dimension of the
parameter space can be found in [49].
Definition 4.13 A topological vector space V on a field K is a vector space for which addition and
scalar multiplication are continuous operations, that is, if {xn}n≥1, {yn}n≥1, x, y ∈ V and {cn}n≥1, c ∈
K with xn −→ x, yn −→ y and cn −→ c, then xn + yn −→ x + y and cnxn −→ cx.
Remark 4.14 In the following results we use some concepts from von Mises calculus like Hadamard
differentiability in topological vector spaces, and we consider the hypothesis that space D[0, 1]d
is a topological vector space. We study the convergence of the sample copula from the results of
convergence of the empirical copula. However, the convergence of the sample copula can be found
in the space C[0, 1]d, because it is a continuous function.
Definition 4.15 Let D and E be metrizable, topological vector spaces, a map ϕ : Dϕ ⊂ D −→ E
is called Hadamard differentiable at θ ∈ Dϕ if there is a continuous linear map ϕ′θ : D −→ E such
that
ϕ(θ + tnhn) − ϕ(θ)
tn
−→ ϕ′θ(h)
if n −→ ∞, for all converging sequences tn −→ 0 and hn −→ h such that θ + tnhn ∈ Dϕ, for all
n ∈ N. We say that ϕ is Hadamard differentiable tangentially to a set D0 ⊂ D by requiring that
every hn −→ h has h ∈ D0.
Definition 4.16 Let (M, ρ) be a given separable, pseudometric space and let (Ω,A, P) be a pro-
bability space. A stochastic process {G(t, ω) : t ∈ M, ω ∈ Ω} is called separable if there exists a
null set N ∈ A and a countable subset G ⊂ M such that, for all ω < N and t ∈ M, there exists a
sequence tn ∈ G, with tn −→ t and G(tn, ω) −→ G(t, ω).
Theorem 4.17 Let (M, ρ) be a given separable, pseudometric space and let (Ω,A, P) be a proba-
bility space. If there is a set A ∈ A with probability zero such that for ω < A, G(t, ω) is continuous
in t, then {G(t, ω) : t ∈ M, ω ∈ Ω} is separable, and G ⊂ M can be taken as any countable dense
subset ofM.
Theorem 4.18 (Delta Method) LetD and E be metrizable, topological vector spaces, let ϕ : Dϕ ⊂
D −→ E be Hadamard differentiable at θ tangentially to D0. Let Xn : Ωn −→ Dϕ be maps with
rn(Xn − θn)
D−→ X for some sequence of constants rn −→ ∞, where X is separable and takes its
values in D0, and θn −→ θ, then rn(ϕ(Xn) − ϕ(θn))
D−→ ϕ′θ(X).
110
The following theorem is given in [13], in this result the convergence of the empirical process
associated to the empirical copula to a Gaussian process is studied. This theorem is the main result
to obtain in the next section the convergence of the process associated to the sample copula, also
from this result we can extend the Theorem 4.12 to get weak convergence without the hypothesis
of independence, with some restrictions.
Theorem 4.19 Let H be a bivariate distribution function with continuous marginal distribution
functions and associated copula function C with continuous partial derivatives. Then the empirical
copula process
Zn =
√
n(Cn −C)
with Cn the empirical copula, converge weakly to the centered Gaussian process GC in l∞([0, 1]2),
the set of all uniformly bounded functions from I2 = [0, 1]2 to R. The limiting Gaussian process
can be written as
Gc(u, v) = BC(u, v) − ∂1C(u, v)BC(u, 1) − ∂2C(u, v)BC(1, v),
where BC is the Brownian sheet on I2 = [0, 1]2 with covariance function
E(BC(u, v) · BC(u′, v′)) = C(u ∧ u′, v ∧ v′) −C(u, v)C(u′, v′).
4.2 Weak convergence of sample copula process
In this section the sample copula is represented as a linear functional evaluated in the empirical
copula, in order to apply the results described in the previous section to get the weak convergence
of the process associated with the sample copula. In this way, the weak convergence theorem for
the empirical copula can be extended for the sample copula.
We begin this section with a simple result which allows us the study of subcopulas with restricted
domains, because later we will consider the empirical copula with a restricted domain in the des-
cription of the sample copula.
Lemma 4.20 Let C′ : S 1×S 2 −→ [0, 1] be a subcopula, let Ŝ 1 ⊂ S 1 and Ŝ 2 ⊂ S 2 such as 0, 1 ∈ Ŝ 1
and 0, 1 ∈ Ŝ 2, then Ĉ = C′|Ŝ 1×Ŝ 2
, with Ĉ(u, v) = C(u, v) for all (u, v) ∈ Ŝ 1 × Ŝ 2 ⊂ S 1 × S 2, is a
subcopula.
Proof: We verified that Ĉ satisfies the subcopulas properties,
1. Dom(Ĉ) = Ŝ 1 × Ŝ 2 ⊂ [0, 1]2 and 0, 1 ∈ Ŝ 1 ∩ Ŝ 2.
111
2. Let u ∈ Ŝ 1 and v ∈ Ŝ 2, then
Ĉ(0, v) = C′(0, v) = 0
Ĉ(u, 0) = C′(u, 0) = 0.
Let a1, a2 ∈ Ŝ 1 and b1, b2 ∈ Ŝ 2, with a1 ≤ a2 and b1 ≤ b2, then
VolĈ([a1, a2] × [b1, b2]) = Ĉ(a2, b2) − Ĉ(a1, b2) − Ĉ(a2, b1) + Ĉ(a1, b1)
= C′(a2, b2) −C′(a1, b2) −C′(a2, b1) +C′(a1, b1)
= VolC′([a1, a2] × [b1, b2])
≥ 0.
3. Let u ∈ Ŝ 1 and v ∈ Ŝ 2, then
Ĉ(u, 1) = C′(u, 1) = u
Ĉ(1, v) = C′(1, v) = v.
Therefore Ĉ is a subcopula.
Remark 4.21 In the following results we use the Nelsen’s notation [37]: For (a, b) ∈ I2 = [0, 1]2
and a subcopula C′ with domain S 1 × S 2, let a1 and a2 be, respectively, the greatest and least
elements of S 1 that satisfy a1 ≤ a ≤ a2; and let b1 and b2 be, respectively, the greatest and least
elements of S 2 that satisfy b1 ≤ b ≤ b2. If a ∈ S 1, then a1 = a = a2, and if b ∈ S 2, then b1 = b = b2.
Let
λ1(a, b) = λ1 =
{
(a − a1)/(a2 − a1) if a1 < a2
1 if a1 = a2
and
µ1(a, b) = µ1 =
{
(b − b1)/(b2 − b1) if b1 < b2
1 if b1 = b2.
If we define
C(a, b) = (1 − λ1)(1 − µ1)C′(a1, b1) + (1 − λ1)µ1C
′(a1, b2) + λ1(1 − µ1)C′(a2, b1) + λ1µ1C
′(a2, b2)
then C is a copula that extends the subcopula C′.
Also, we considered Cn and Cn
m (m ≥ 2), respectively, the empirical copula and the sample copula,
build from the modified sample of a sample of size n from a copula C.
112
In the following results the sample copula is represented like a linear functional evaluated in the
empirical copula and we proof that this functional is Hadamard differentiable. These properties
are required in order to use the delta method for the study of weak convergence of the process
associated to the sample copula.
Lemma 4.22 Let m ≥ 2 and ϕm : D[0, 1]2 −→ l∞([0, 1]2), with
ϕm(H) =
m
∑
j=1
m
∑
i=1
[
1( i−1
m
, i
m ]×
(
j−1
m
,
j
m
](1 − λ1)(1 − µ1)H
(
i − 1
m
,
j − 1
m
)
+ (1 − λ1)µ1H
(
i − 1
m
,
j
m
)
+ λ1(1 − µ1)H
(
i
m
,
j − 1
m
)
+ λ1µ1H
(
i
m
,
j
m
)
]
then
ϕm(Cn) = Cn
m.
Proof: We consider (a, b) ∈ I2 = [0, 1]2 with (a, b) ∈ ((i − 1)/m, i/m] × (( j − 1)/m, j/m
]
and let λ
be the Lebesgue’s measure in B([0, 1]2). We define
C(a, b) = (1 − λ1)(1 − µ1)Ĉn
(
i − 1
m
,
j − 1
m
)
+ (1 − λ1)µ1Ĉn
(
i − 1
m
,
j
m
)
+λ1(1 − µ1)Ĉn
(
i
m
,
j − 1
m
)
+ λ1µ1Ĉn
(
i
m
,
j
m
)
where λ1 = λ1(a, b) = (a−(i−1)/m)/(i/m−(i−1)/m), µ1 = µ1(a, b) = (b−( j−1)/m)/( j/m−( j−1)/m)
and Ĉn is the subcopula obtained from the empirical copula Cn restricted to a domain consisting
of points of the form (i/m, j/m), with i, j ∈ {1, · · · ,m}, from the Lemma 4.20, Ĉ is a subcopula
and therefore C is a copula. We will denote Ĉn by Cn and we calculate with this definition the
C-volumen of the rectangle ((i − 1)/m, a] × (( j − 1)/m, b
]
,
VolC
((
i − 1
m
, a
]
×
(
j − 1
m
, b
])
= C(a, b) −C
(
a,
j − 1
m
)
−C
(
i − 1
m
, b
)
+C
(
i − 1
m
,
j − 1
m
)
,
where
C
(
a,
j − 1
m
)
= (1 − λ1)Cn
(
i − 1
m
,
j − 1
m
)
+ λ1Cn
(
i
m
,
j − 1
m
)
,
C
(
i − 1
m
, b
)
= (1 − µ1)Cn
(
i − 1
m
,
j − 1
m
)
+ µ1Cn
(
i − 1
m
,
j
m
)
,
113
and
C
(
i − 1
m
,
j − 1
m
)
= Cn
(
i − 1
m
,
j − 1
m
)
,
we have that
C(a, b) −C
(
a,
j−1
m
)
−C
(
i−1
m
, b
)
+C
(
i−1
m
,
j−1
m
)
=
[
(1 − λ1)(1 − µ1) − (1 − λ1) − (1 − µ1) + 1
]
Cn
(
i−1
m
,
j−1
m
)
+
[
(1 − λ1)µ1 − µ1
]
Cn
(
i−1
m
,
j
m
)
+
[
λ1(1 − µ1) − λ1
]
Cn
(
i
m
,
j−1
m
)
+ λ1µ1Cn
(
i
m
,
j
m
)
= λ1µ1
[
Cn
(
i
m
,
j
m
)
−Cn
(
i−1
m
,
j
m
)
−Cn
(
i
m
,
j−1
m
)
+Cn
(
i−1
m
,
j−1
m
)]
.
Therefore
VolC
((
i − 1
m
, a
]
×
(
j − 1
m
, b
])
= λ1µ1VolCn
((
i − 1
m
,
i
m
]
×
(
j − 1
m
,
j
m
])
= λ1µ1si j
=
λ
((
i−1
m
, a
]
×
(
j−1
m
, b
])
λ
((
i−1
m
, i
m
]
×
(
j−1
m
,
j
m
]) si j
= VolCn
m
((
i − 1
m
, a
]
×
(
j − 1
m
, b
])
.
Now, we calculate the C-volume of the box ((i − 1)/m, i/m] × (( j − 1)/m, b],
VolC
((
i − 1
m
,
i
m
]
×
(
j − 1
m
, b
])
= C
(
i
m
, b
)
−C
(
i − 1
m
, b
)
−C
(
i
m
,
j − 1
m
)
+C
(
i − 1
m
,
j − 1
m
)
where
C
(
i
m
, b
)
=







j
m
− b
j
m
− j−1
m







Cn
(
i
m
,
j − 1
m
)
+







b − j−1
m
j
m
− j−1
m







Cn
(
i
m
,
j
m
)
,
C
(
i − 1
m
, b
)
=







j
m
− b
j
m
− j−1
m







Cn
(
i − 1
m
,
j − 1
m
)
+







b − j−1
m
j
m
− j−1
m







Cn
(
i − 1
m
,
j
m
)
,
then
VolC
((
i − 1
m
,
i
m
]
×
(
j − 1
m
, b
])
= C
(
i
m
, b
)
−C
(
i − 1
m
, b
)
−C
(
i
m
,
j − 1
m
)
+C
(
i − 1
m
,
j − 1
m
)
114
=














j
m
− b
j
m
− j−1
m







− 1







Cn
(
i
m
,
j − 1
m
)
+







j−1
m
− b
j
m
− j−1
m







Cn
(
i
m
,
j
m
)
+














b − j
m
j
m
− j−1
m







+ 1







Cn
(
i − 1
m
,
j − 1
m
)
−







j−1
m
− b
j
m
− j−1
m







Cn
(
i − 1
m
,
j
m
)
=







b − j−1
m
j
m
− j−1
m







[
Cn
(
i
m
,
j
m
)
−Cn
(
i
m
,
j − 1
m
)
−Cn
(
i − 1
m
,
j
m
)
+Cn
(
i − 1
m
,
j − 1
m
)]
=







b − j−1
m
j
m
− j−1
m













i
m
− i−1
m
i
m
− i−1
m






VCn
((
i − 1
m
,
i
m
]
×
(
j − 1
m
,
j
m
])
=
λ
((
i−1
m
, i
m
]
×
(
j−1
m
, b
])
λ
((
i−1
m
, i
m
]
×
(
j−1
m
,
j
m
]) si j
= VolCn
m
((
i − 1
m
,
i
m
]
×
(
j − 1
m
, b
])
.
Finally, we calculate the C-volume of the box ((i − 1)/m, a] × (( j − 1)/m, j/m],
VolC
((
i − 1
m
, a
]
×
(
j − 1
m
,
j
m
])
= C
(
a,
j
m
)
−C
(
i − 1
m
,
j
m
)
−C
(
a,
j − 1
m
)
+C
(
i − 1
m
,
j − 1
m
)
where
C
(
a,
j
m
)
=






i
m
− a
i
m
− i−1
m






Cn
(
i − 1
m
,
j
m
)
+






a − i−1
m
i
m
− i−1
m






Cn
(
i
m
,
j
m
)
,
C
(
a,
j − 1
m
)
=






i
m
− a
i
m
− i−1
m






Cn
(
i − 1
m
,
j − 1
m
)
+






a − i−1
m
i
m
− i−1
m






Cn
(
i
m
,
j − 1
m
)
,
then
VolC
((
i − 1
m
, a
]
×
(
j − 1
m
,
j
m
])
= C
(
a,
j
m
)
−C
(
i − 1
m
,
j
m
)
−C
(
a,
j − 1
m
)
+C
(
i − 1
m
,
j − 1
m
)
=












i
m
− a
i
m
− i−1
m






− 1






Cn
(
i − 1
m
,
j
m
)
+






a − i−1
m
i
m
− i−1
m






Cn
(
i
m
,
j
m
)
+












a − i
m
i
m
− i−1
m






+ 1






Cn
(
i − 1
m
,
j − 1
m
)
−






a − i−1
m
i
m
− i−1
m






Cn
(
i
m
,
j − 1
m
)
115
=






a − i−1
m
i
m
− i−1
m






[
Cn
(
i
m
,
j
m
)
−Cn
(
i
m
,
j − 1
m
)
−Cn
(
i − 1
m
,
j
m
)
+Cn
(
i − 1
m
,
j − 1
m
)]
=






a − i−1
m
i
m
− i−1
m













j
m
− i−1
m
j
m
− j−1
m







VCn
((
i − 1
m
,
i
m
]
×
(
j − 1
m
,
j
m
])
=
λ
((
i−1
m
, a
]
×
(
j−1
m
,
j
m
])
λ
((
i−1
m
, i
m
]
×
(
j−1
m
,
j
m
]) si j
= VolCn
m
((
i − 1
m
, a
]
×
(
j − 1
m
, b
])
.
Because any box can be written as the union of boxes considered in the previous cases we can
conclude that
ϕm(Cn) = Cn
m.

Lemma 4.23 Let m ≥ 2 and ϕm : D[0, 1]2 −→ l∞([0, 1]2), with
ϕm(H) =
m
∑
j=1
m
∑
i=1
[
1( i−1
m
, i
m ]×
(
j−1
m
,
j
m
](1 − λ1)(1 − µ1)H
(
i − 1
m
,
j − 1
m
)
+ (1 − λ1)µ1H
(
i − 1
m
,
j
m
)
+ λ1(1 − µ1)H
(
i
m
,
j − 1
m
)
+ λ1µ1H
(
i
m
,
j
m
)
]
then ϕm is Hadamard differentiable in every H ∈ D[0, 1]2.
Proof: Let {hn}n≥1 ⊂ D[0, 1]2, H ∈ D[0, 1]2, and {tn}n≥1 ⊂ R such as hn −→ h, with h ∈ D[0, 1]2,
and tn −→ 0. It holds
lim
n−→∞
ϕm(H + tnhn) − ϕm(H)
tn
= lim
n−→∞








m
∑
j=1
m
∑
i=1
[
1( i−1
m
, i
m ]×
(
j−1
m
,
j
m
](1 − λ1)(1 − µ1)
(
H
(
i − 1
m
,
j − 1
m
)
+tnhn
(
i − 1
n
,
j − 1
m
))
+ (1 − λ1)µ1
(
H
(
i − 1
m
,
j
m
)
+ tnhn
(
i − 1
m
,
j
m
))
+λ1(1 − µ1)
(
H
(
i
m
,
j − 1
m
)
+ tnhn
(
i
m
,
j − 1
m
))
+λ1µ1
(
H
(
i
m
,
j
m
)
+ tnhn
(
i
m
,
j
m
))]
− ϕm(H)
]
/tn
116
= lim
n−→∞








m
∑
j=1
m
∑
i=1
[
1( i−1
m
, i
m ]×
(
j−1
m
,
j
m
](1 − λ1)(1 − µ1)hn
(
i − 1
n
,
j − 1
m
)
+(1 − λ1)µ1hn
(
i − 1
m
,
j
m
)
+ λ1(1 − µ1)hn
(
i
m
,
j − 1
m
)
+λ1µ1hn
(
i
m
,
j
m
)]]
=
m
∑
j=1
m
∑
i=1
[
1( i−1
m
, i
m ]×
(
j−1
m
,
j
m
](1 − λ1)(1 − µ1)h
(
i − 1
n
,
j − 1
m
)
+(1 − λ1)µ1h
(
i − 1
m
,
j
m
)
+ λ1(1 − µ1)h
(
i
m
,
j − 1
m
)
+λ1µ1h
(
i
m
,
j
m
)]
= ϕm(h).
Let c ∈ R and h1, h2 ∈ D[0, 1]2, we have
ϕm(ch1 + h2) =
m
∑
j=1
m
∑
i=1
[
1( i−1
m
, i
m ]×
(
j−1
m
,
j
m
](1 − λ1)(1 − µ1)(ch1 + h2)
(
i − 1
n
,
j − 1
m
)
+(1 − λ1)µ1(ch1 + h2)
(
i − 1
m
,
j
m
)
+ λ1(1 − µ1)(ch1 + h2)
(
i
m
,
j − 1
m
)
+λ1µ1(ch1 + h2)
(
i
m
,
j
m
)]
= c
m
∑
j=1
m
∑
i=1
[
1( i−1
m
, i
m ]×
(
j−1
m
,
j
m
](1 − λ1)(1 − µ1)h1
(
i − 1
n
,
j − 1
m
)
+(1 − λ1)µ1h1
(
i − 1
m
,
j
m
)
+ λ1(1 − µ1)h1
(
i
m
,
j − 1
m
)
+λ1µ1h1
(
i
m
,
j
m
)]
+
m
∑
j=1
m
∑
i=1
[
1( i−1
m
, i
m ]×
(
j−1
m
,
j
m
](1 − λ1)(1 − µ1)h2
(
i − 1
n
,
j − 1
m
)
+(1 − λ1)µ1h2
(
i − 1
m
,
j
m
)
+ λ1(1 − µ1)h2
(
i
m
,
j − 1
m
)
+λ1µ1h2
(
i
m
,
j
m
)]
= cϕm(h1) + ϕm(h2)
and ϕm is linear.
117
Let {hn}n≥1 ⊂ D[0, 1]2 with hn −→ h ∈ D[0, 1]2, it holds
lim
n−→∞
ϕm(hn) = lim
n−→∞
m
∑
j=1
m
∑
i=1
[
1( i−1
m
, i
m ]×
(
j−1
m
,
j
m
](1 − λ1)(1 − µ1)hn
(
i − 1
n
,
j − 1
m
)
+(1 − λ1)µ1hn
(
i − 1
m
,
j
m
)
+ λ1(1 − µ1)hn
(
i
m
,
j − 1
m
)
+λ1µ1hn
(
i
m
,
j
m
)]
=
m
∑
j=1
m
∑
i=1
[
1( i−1
m
, i
m ]×
(
j−1
m
,
j
m
](1 − λ1)(1 − µ1)h
(
i − 1
n
,
j − 1
m
)
+(1 − λ1)µ1h
(
i − 1
m
,
j
m
)
+ λ1(1 − µ1)h
(
i
m
,
j − 1
m
)
+λ1µ1h
(
i
m
,
j
m
)]
= ϕm(h)
and ϕm is continuous.
Therefore we can define the Hadamard derivative in H of ϕm as ϕ′H(h) = ϕm(h), for all h ∈ D[0, 1]2.

Remark 4.24 The functional defined in Lemma 4.22 applied to C is exactly the checkerboard
copula of order m defined in [33], and the linear B-spline copula defined in [43], in the case m
equals to n. In Chapter 2 we can see that we obtain a better approximation to the real copula if we
consider m smaller than n.
Remark 4.25 Because in the proof of Lemma 4.23 the relations holds for every sequence {hn}n≥1 ⊂
D[0, 1]2 and h ∈ D[0, 1]2 such that hn −→ h, particularly we can take h ∈ C[0, 1]2 ⊂ D[0, 1]2, that
is, ϕm is Hadamard differentiable in every H ∈ D[0, 1]2 tangentially at D0 = C[0, 1]2.
The following theorem is the main result of this chapter, since it provides the weak convergence
of the process associated with the sample copula. It is important to notes that the proof of the
following theorem uses the convergence of the process associated to the empirical copula given in
the Theorem 4.19.
Theorem 4.26 Let m ≥ 2 be fixed, let C be a copula with continuous partial derivatives and let
Cn
m be the sample copula build from the modified sample of a sample of size n from C. Let {rn}n≥1
be a increasing sequence such as rn −→ ∞ and m divides rn, for all n ∈ N, then
√
rn
(
Crn
m − ϕm(C)
) D−→ ϕ′C(GC) = ϕm(GC)
118
where ϕm is defined in the Lemma 4.23, and ϕ′CGC) = ϕm(GC) = G is defined by
G =
m
∑
j=1
m
∑
i=1
[
1( i−1
m
, i
m ]×
(
j−1
m
,
j
m
](1 − λ1)(1 − µ1)GC
(
i − 1
m
,
j − 1
m
)
+(1 − λ1)µ1GC
(
i − 1
m
,
j
m
)
+ λ1(1 − µ1)GC
(
i
m
,
j − 1
m
)
+λ1µ1GC
(
i
m
,
j
m
)]
.
Proof: Based on the Theorem 4.19 we have that
√
n (Cn −C)
D−→ GC,
if we define rn = mn, for all n ∈ N, then the above relation implies that
√
rn
(
Crn
−C
) D−→ GC,
when n −→ ∞, the Lemma 4.23 and Remark 4.25 imply that ϕm is Hadamard differentiable at
every H ∈ D[0, 1]2 tangentially at C[0, 1]2; we can consider, by Theorem 4.17, that the processes
GC ∈ C[0, 1]2 is a separable stochastic process, because it is continuous almost surely respect to
a probability measure P. Applying the Delta Method (Theorem 4.18) with Dϕ = D = D[0, 1]2,
E = l∞([0, 1]2), θrn
= C, for all n ∈ N, θ = C, and D0 = C[0, 1]2, we have that
√
rn
(
ϕm(Crn
) − ϕm(C)
) D−→ ϕ′C(GC) = ϕm(GC)
and, by the Lemma 4.22, it is satisfied that ϕm(Crn
) = C
rn
m , therefore
√
rn
(
Crn
m − ϕm(C)
) D−→ ϕ′C(GC) = ϕm(GC).

In the following remark we make observations about the parameters in the Gaussian process.
Remark 4.27 Let (a, b) ∈ I2 = [0, 1]2 with (a, b) ∈ (i − 1, i/m] × ( j − 1, j/m], because the linear
combination of the components of a Gaussian vector have normal distribution and the process GC
is centered [29, Definition 16.1], we have that ϕ′C(GC)(a, b) = ϕm(GC)(a, b) = G(a, b) is normally
119
distributed with mean equal to zero and variance given by (using the values λ1(a, b) and µ1(a, b),
given on the Remark 4.21, and the results from Theorem 4.19)
Var(G(a, b)) = (1 − λ1)2(1 − µ1)2Var
(
GC
(
i − 1
m
,
j − 1
m
))
+ (1 − λ1)2µ2
1Var
(
GC
(
i − 1
m
,
j
m
))
+λ2
1(1 − µ1)2Var
(
GC
(
i
m
,
j − 1
m
))
+ λ2
1µ
2
1Var
(
GC
(
i
m
,
j
m
))
+2(1 − λ1)2µ1(1 − µ1)E
(
GC
(
i − 1
m
,
j − 1
m
)
· GC
(
i − 1
m
,
j
m
))
+2λ1(1 − λ1)(1 − µ1)2E
(
GC
(
i − 1
m
,
j − 1
m
)
· GC
(
i
m
,
j − 1
m
))
+2λ1(1 − λ1)µ1(1 − µ1)E
(
GC
(
i − 1
m
,
j − 1
m
)
· GC
(
i
m
,
j
m
)
)
+2λ1(1 − λ1)µ1(1 − µ1)E
(
GC
(
i − 1
m
,
j
m
)
· GC
(
i
m
,
j − 1
m
))
+2λ1(1 − λ1)µ2
1E
(
GC
(
i − 1
m
,
j
m
)
· GC
(
i
m
,
j
m
)
)
+2λ2
1µ1(1 − µ1)E
(
GC
(
i
m
,
j − 1
m
)
· GC
(
i
m
,
j
m
)
)
with
E
[
GC(u, v) · GC(u′, v′)
]
= E [(BC(u, v) − ∂1C(u, v)BC(u, 1) − ∂2C(u, v)BC(1, v))
·(BC(u′, v′) − ∂1C(u′, v′)BC(u′, 1) − ∂2C(u′, v′)BC(1, v′))
]
= E
[
BC(u, v)BC(u′, v′)
] − E
[
BC(u, v)∂1C(u′, v′)BC(u′, 1)
]
−E
[
BC(u, v)∂2C(u′, v′)BC(1, v′)
] − E
[
∂1C(u, v)BC(u, 1)BC(u′, v′)
]
+E
[
∂1C(u, v)BC(u, 1)∂1C(u′, v′)BC(u′, 1)
]
+E
[
∂1C(u, v)BC(u, 1)∂2C(u′, v′)BC(1, v′)
]
−E
[
∂2C(u, v)BC(1, v)BC(u′, v′)
]
+E
[
∂2C(u, v)BC(1, v)∂1C(u′, v′)BC(u′, 1)
]
+E
[
∂2C(u, v)BC(1, v)∂2C(u′, v′)BC(1, v′)
]
= C(u ∧ u′, v ∧ v′) −C(u, v)C(u′, v′)
−∂1C(u′, v′)
[
C(u ∧ u′, v) − u′C(u, v)
]
−∂2C(u′, v′)
[
C(u, v ∧ v′) − v′C(u, v)
]
120
−∂1C(u, v)
[
C(u ∧ u′, v′) − uC(u′, v′)
]
+∂1C(u, v)∂1C(u′, v′)
[
u ∧ u′ − uu′
]
+∂1C(u, v)∂2C(u′, v′)
[
C(u, v′) − uv′
]
−∂2C(u, v)
[
C(u′, v ∧ v′) − vC(u′, v′)
]
+∂2C(u, v)∂1C(u′, v′)
[
C(u′, v) − vu′
]
∂2C(u, v)∂2C(u′, v′)
[
v ∧ v′ − vv′
]
and
Var(GC(u, v)) = C(u, v) −C(u, v)2 − ∂1C(u, v) [C(u, v) −C(u, v)u]
−∂2C(u, v) [C(u, v) −C(u, v)v] − ∂1C(u, v) [C(u, v) −C(u, v)u]
+∂1C(u, v)∂1C(u, v)
(
u − u2
)
+ ∂2C(u, v)∂1C(u, v) [C(u, v) − uv]
−∂2C(u, v) [C(u, v) − vC(u, v)] + ∂2C(u, v)∂1C(u, v) [C(u, v) − uv]
+∂2C(u, v)∂2C(u, v)
(
v − v2
)
.
Remark 4.28 In the case where the copula C is equal to the product copula, for all (a, b) ∈ I2 =
[0, 1]2, using Nelsen’s notation given in the Remark 4.21 and taking a1 = (i − 1)/m, a2 = i/m, b1 =
( j − 1)/m and b2 = j/m, we have
C(a, b) = (1 − λ1)(1 − µ1)Π(a1, b1) + (1 − λ1)µ1Π(a1, b2) + λ1(1 − µ1)Π(a2, b1) + λ1µ1Π(a2, b2)
=
(
a2 − a
a2 − a1
) (
b2 − b
b2 − b1
)
a1b1 +
(
a2 − a
a2 − a1
) (
b − b1
b2 − b1
)
a1b2
+
(
a − a1
a2 − a1
) (
b2 − b
b2 − b1
)
a2b1 +
(
a − a1
a2 − a1
) (
b − b1
b2 − b1
)
a2b2
(
a2 − a
a2 − a1
)
a1
[(
b2 − b
b2 − b1
) (
b − b1
b2 − b1
)
b2
]
+
(
a − a1
a2 − a1
)
a2
[(
b2 − b
b2 − b1
) (
b − b1
b2 − b1
)
b2
]
=
(
a2 − a
a2 − a1
)
a1b +
(
a − a1
a2 − a1
)
a2b
=
a2 − a1
a2 − a2
ab
= ab
= Π(a, b).
Therefore the conclusion of Theorem 4.26, can we written as
√
rn
(
Crn
m − Π
) D−→ ϕ′H(GC) = ϕm(GC).
121
In this case, the covariance and variance of Z = GC processes is given by
Cov(Z(s, t),Z(u, v)) = (s ∧ u − su)(t ∧ v − tv)
and
Var(Z(s, t)) = (s − s2)(t − t2).
Let (a, b) ∈ I2 = [0, 1]2 with (a, b) ∈ (i − 1, i/m] × ( j − 1, j/m], for the calculus of variance of the
ϕm(Z)(a, b), normally distributed with mean equal to zero, we have the following relations
V1 = Var
(
(1 − λ1)(1 − µ1)Z
(
i − 1
n
,
j − 1
m
))
= (1 − λ1)2(1 − µ1)2






i − 1
m
−
(
i − 1
m
)2











j − 1
m
−
(
j − 1
m
)2





,
V2 = Var
(
(1 − λ1)µ1Z
(
i − 1
n
,
j
m
))
= (1 − λ1)2µ2
1






i − 1
m
−
(
i − 1
m
)2





(
j
m
−
(
j
m
)2
)
,
V3 = Var
(
λ1(1 − µ1)Z
(
i
n
,
j − 1
m
))
= λ2
1(1 − µ1)2
(
i
m
−
(
i
m
)2
) 





j − 1
m
−
(
j − 1
m
)2





,
V4 = Var
(
λ1µ1Z
(
i
n
,
j
m
))
= λ2
1µ
2
1
(
i
m
−
(
i
m
)2
) (
j
m
−
(
j
m
)2
)
,
C12 = Cov
(
(1 − λ1)(1 − µ1)Z
(
i − 1
m
,
j − 1
m
)
, (1 − λ1)µ1Z
(
i − 1
m
,
j
m
))
= (1 − λ1)2(1 − µ1)µ1






i − 1
m
−
(
i − 1
m
)2





(
j − 1
m
− j( j − 1)
m2
)
,
C13 = Cov
(
(1 − λ1)(1 − µ1)Z
(
i − 1
m
,
j − 1
m
)
, λ1(1 − µ1)Z
(
i
m
,
j − 1
m
))
= λ1(1 − λ1)(1 − µ1)2
(
i − 1
m
− i(i − 1)
m2
) 





j − 1
m
−
(
j − 1
m
)2





,
C14 = Cov
(
(1 − λ1)(1 − µ1)Z
(
i − 1
m
,
j − 1
m
)
, λ1µ1Z
(
i
m
,
j
m
)
)
122
= λ1(1 − λ1)µ1(1 − µ1)
(
i − 1
m
− i(i − 1)
m2
) (
j − 1
m
− j( j − 1)
m2
)
,
C23 = Cov
(
(1 − λ1)µ1Z
(
i − 1
m
,
j
m
)
, λ1(1 − µ1)Z
(
i
m
,
j − 1
m
))
= λ1(1 − λ1)µ1(1 − µ1)
(
i − 1
m
− i(i − 1)
m2
) (
j − 1
m
− j( j − 1)
m2
)
,
C24 = Cov
(
(1 − λ1)µ1Z
(
i − 1
m
,
j
m
)
, λ1µ1Z
(
i
m
,
j
m
)
)
= λ1(1 − λ1)µ2
1
(
i − 1
m
− i(i − 1)
m2
) (
j
m
−
(
j
m
)2
)
,
C34 = Cov
(
λ1(1 − µ1)Z
(
i
m
,
j − 1
m
)
, λ1µ1Z
(
i
m
,
j
m
)
)
= λ2
1µ1(1 − µ1)
(
i
m
−
(
i
m
)2
) (
j − 1
m
− j( j − 1)
m2
)
,
and
Var(ϕm(Z)(a, b)) = V1 + V2 + V3 + V4 + 2 (C12 +C13 +C14 +C23 +C24 +C34) .
From [40], and using similar calculations as above, we can extend the convergence results of the
Theorem 4.26 to higher dimensions d > 2. This is given in the following theorem:
Theorem 4.29 Let d > 2, m ≥ 2, let C be a d-copula with continuous partial derivatives and let
Cn
m be the sample copula build from the modified sample of a sample of size n from C. Let {rn}n≥1
be a increasing sequence such as rn −→ ∞ and m divides rn, for all n ∈ N, then
√
rn
(
Crn
m − ϕm(C)
) D−→ ϕ′C(GC) = ϕm(GC)
where
GC = BC(u1, · · · , ud) −
d
∑
i=1
∂iC(u1, · · · , ud)BC(1, 1, · · · , ui, · · · , 1)
and BC is a Gaussian process with
E(BC(u1, · · · , ud) · BC(u′1, · · · , u′d)) = C(u1 ∧ u′1, · · · , ud ∧ u′d) −C(u1, · · · , ud)C(u′1, · · · , u′d).
In the following section we given the results of perform several simulations of the associated
process of the sample copula given in Theorem 4.26 to exemplify its convergence.
123
4.3 Simulation study: Gaussian process
In the next tables we present the results when we perform 1, 000 iterations from samples of size
50, 000 for the process associated to some family of copulas with continuous partial derivatives.
The tables show the results when we repeat these simulations 100 times. The column θ indicates
the parameter value, the column ρ indicates the value of the Spearman’s Rho associated to the
parameter of the copula, the column Point gives the point of evaluation of the process, the column
M. means is the mean of the means of the 100 simulations of the process (since the processes is a
centered one, this value most be close to zero), M. var denotes the mean of the variances obtained
form the 100 repetitions, Real var. denotes the real variance calculated using Remark 4.27, the
firsts three columns 0.01, 0.05 and 0.10 indicates the number of rejections of normality in the
100 simulations at α−level using the Anderson-Darling’s test, and the second three columns 0.01,
0.05 and 0.10 indicates the number of rejections of normality in the 100 simulations at α−level
using the Shapiro-Wilk’s test, the column UAD indicates the p-value of a Kolmogorov-Smirnov’s
uniformity test applied to the p-values of Anderson-Darling’s tests, and the column USW indicates
the p-value of a Kolmogorov-Smirnov’s uniformity test applied to the p-values of Shapiro-Wilk’s
tests. In all cases we consider the parameter m of the sample copula equals to 4.
θ ρ Point M. means M. var. Real var. 0.01 0.05 0.10 0.01 0.05 0.10 UAD USW
20 0.9578 (0.8,0.9) -0.00010 0.001643 0.001638 1 4 11 1 5 14 0.00032 0.02205
20 0.9578 (0.1,0.2) -0.00023 0.00163 0.00163 2 10 16 3 8 15 0.00036 0.01440
20 0.9578 (0.4,0.6) -0.00004 0.00220 0.00220 3 11 15 0 5 11 0.34306 0.38131
-20 -0.9578 (0.1,0.9) 0.00010 0.00040 0.00040 2 9 16 0 6 14 0.00299 0.00692
-20 -0.9578 (0.8,0.2) 0.00034 0.00654 0.00655 2 7 12 3 7 10 0.01939 0.12012
-20 -0.9578 (0.4,0.6) 0.00046 0.00252 0.00252 0 4 11 3 8 12 0.74344 0.98920
5 0.6434 (0.1,0.2) -0.00027 0.00378 0.00377 0 4 11 0 5 8 0.06614 0.75172
5 0.6434 (0.8,0.9) -0.00001 0.00377 0.00377 1 8 14 2 9 13 0.13685 0.27477
5 0.6434 (0.4,0.6) -0.00047 0.01665 0.01672 0 2 9 0 4 7 0.01777 0.57689
-5 -0.6434 (0.1,0.9) -0.00002 0.00093 0.00094 2 6 17 1 7 14 0.00298 0.03665
-5 -0.6434 (0.8,0.2) 0.00005 0.01518 0.01508 1 5 10 2 9 16 0.04696 0.15866
-5 -0.6434 (0.4,0.6) 0.00040 0.01919 0.01916 1 2 4 1 3 5 0.017208 0.02306
Table 4.1 Simulations results (Frank Copula).
124
θ ρ Point M. means M. var. Real var. 0.01 0.05 0.10 0.01 0.05 0.10 UAD USW
1.1 0.1341 (0.8,0.9) 0.00009 0.00400 0.00400 1 3 7 1 3 5 0.13436 0.59930
1.1 0.1341 (0.1,0.2) -0.00011 0.00383 0.00384 2 5 12 1 3 6 0.07756 0.24805
1.1 0.1341 (0.4,0.6) -0.00139 0.03150 0.03164 2 5 10 0 7 11 0.44923 0.48465
3 0.8481 (0.8,0.81) -0.00027 0.00889 0.00891 1 7 20 3 10 19 0.00000 0.00000
3 0.8481 (0.1,0.15) -0.00016 0.00171 0.00171 1 4 13 0 5 13 0.04158 0.44648
3 0.8481 (0.45,0.5) 0.00029 0.02529 0.02537 3 5 11 2 6 8 0.99269 0.41925
8 0.9773 (0.8,0.81) -0.00035 0.00346 0.00345 2 9 17 1 8 18 0.00000 0.01543
8 0.9773 (0.1,0.13) -0.00005 0.00058 0.00058 2 8 18 3 8 16 0.00036 0.02517
8 0.9773 (0.45,0.48) -0.00078 0.00787 0.00785 2 6 10 2 4 6 0.67848 0.46945
20 0.9963 (0.8,0.81) -0.00056 0.00138 0.00138 6 23 37 4 15 21 0.00000 0.00000
20 0.9963 (0.1,0.11) -0.00018 0.00017 0.00017 0 14 31 4 11 19 0.00000 0.00000
20 0.9963 (0.4,0.6) -0.00020 0.00076 0.00076 2 10 26 2 10 20 0.00000 0.00000
Table 4.2 Simulations results (Gumbel Copula).
θ ρ Point M. means M. var. Real var. 0.01 0.05 0.10 0.01 0.05 0.10 UAD USW
-0.9 -0.8978 (0.8,0.3) 0.00005 0.01399 0.01408 2 7 8 2 4 11 0.23918 0.39465
-0.9 -0.8978 (0.8,0.9) 0.00007 0.00078 0.00077 0 6 20 3 10 18 0.00000 0.00000
-0.5 -0.9878 (0.5,0.49) -0.00014 0.05621 0.05624 1 3 8 2 3 7 0.74006 0.51611
-0.1 -0.0787 (0.8,0.9) -0.00022 0.00347 0.00346 1 3 5 1 4 6 0.00465 0.06234
-0.1 -0.0787 (0.8,0.3) -0.00052 0.02112 0.02131 0 2 6 0 3 4 0.78381 0.23228
-0.1 -0.0787 (0.2,0.9) -0.00012 0.00380 0.00378 1 6 11 2 6 11 0.00456 0.06511
5 0.8848 (0.1,0.12) -0.00019 0.00058 0.00058 0 9 12 2 10 19 0.01646 0.16689
5 0.8848 (0.8,0.78) -0.00126 0.1646 0.01647 0 4 11 0 5 13 0.08772 0.10310
5 0.8848 (0.4,0.38) -0.00068 0.00447 0.00446 1 4 8 1 5 10 0.50330 0.8815
20 0.9869 (0.1,0.11) -0.00014 0.00013 0.00013 2 15 30 1 7 20 0.00000 0.00000
20 0.9869 (0.8,0.79) -0.00052 0.00574 0.00573 4 15 16 3 11 18 0.00142 0.05822
20 0.9869 (0.4,0.39) -0.00046 0.00109 0.00109 0 0 8 0 2 6 0.06122 0.67889
Table 4.3 Simulations results (Clayton Copula).
125
θ ρ Point M. means M. var. Real var. 0.01 0.05 0.10 0.01 0.05 0.10 UAD USW
1.1 0.0820 (0.8,0.9) -0.00016 0.00396 0.00393 2 4 7 2 6 9 0.29601 0.37343
1.1 0.0820 (0.1,0.2) -0.00001 0.00371 0.00371 3 6 9 0 7 12 0.32767 0.58562
1.1 0.0820 (0.4,0.6) 0.00008 0.03230 0.03214 3 8 13 1 7 13 0.63409 0.85489
3 0.6993 (0.8,0.79) -0.00035 0.01340 0.01332 0 7 11 1 6 8 0.18609 0.12258
3 0.6993 (0.1,0.12) 0.00014 0.00148 0.00147 2 5 12 3 6 11 0.50472 0.16758
3 0.6993 (0.4,0.41) -0.00065 0.02382 0.02363 2 5 10 1 4 15 0.56857 0.33761
8 0.9310 (0.8,0.79) -0.00047 0.00507 0.00506 2 10 19 1 11 20 0.00000 0.00000
8 0.9310 (0.1,0.12) 0.00006 0.00104 0.00104 1 4 9 1 1 7 0.01724 0.42950
8 0.9310 (0.4,0.41) -0.00058 0.00566 0.00568 0 3 9 1 5 8 0.06168 0.12568
20 0.9860 (0.7,0.7) -0.00064 0.00182 0.00181 2 4 9 1 7 12 0.71708 0.19691
20 0.9860 (0.1,0.1) -0.00012 0.00033 0.00033 4 10 17 4 8 12 0.00094 0.09628
20 0.9860 (0.4,0.4) -0.00040 0.00148 0.00148 2 4 9 1 7 12 0.96922 0.27021
Table 4.4 Simulations results (Joe Copula).
θ ρ Point M. means M. var. Real var. 0.01 0.05 0.10 0.01 0.05 0.10 UAD USW
0.1 -0.6536 (0.8,0.2) 0.00004 0.01436 0.01445 2 2 8 2 7 11 0.14030 0.19872
0.1 -0.6536 (0.1,0.9) 0.00015 0.00089 0.00090 3 7 9 2 5 7 0.48364 0.61248
0.1 -0.6536 (0.4,0.41) 0.00016 0.01803 0.01811 1 7 13 3 8 17 0.32480 0.0568
1.5 0.1344 (0.8,0.9) 0.00007 0.00392 0.00390 1 4 12 2 7 13 0.22429 0.58390
1.5 0.1344 (0.1,0.2) 0.00006 0.00391 0.00390 1 6 11 1 3 13 0.00762 0.58638
1.5 0.1344 (0.4,0.6) 0.00036 0.03165 0.03145 2 5 9 0 5 13 0.62387 0.13984
8 0.6067 (0.8,0.79) -0.00039 0.01646 0.01658 3 10 14 3 8 17 0.22829 0.48114
8 0.6067 (0.1,0.11) 0.00010 0.00113 0.00113 1 5 9 2 5 10 0.53682 0.92631
8 0.6067 (0.4,0.41) -0.00105 0.02172 0.02170 0 3 9 0 4 8 0.05772 0.04771
30 0.8263 (0.8,0.79) -0.00065 0.01200 0.01199 1 8 17 0 7 13 0.02233 0.09471
30 0.8263 (0.1,0.11) -0.00009 0.00082 0.00082 0 2 10 2 5 11 0.40344 0.60288
30 0.8263 (0.4,0.41) -0.00090 0.01093 0.01095 5 9 11 3 10 12 0.61730 0.82075
Table 4.5 Simulations results (Plackett Copula).
Point M. means M. var. Real var. 0.01 0.05 0.10 0.01 0.05 0.10 UAD USW
(0.8,0.9) -0.00003 0.00366 0.00360 0 5 13 1 9 14 0.18762 0.91708
(0.1,0.2) 0.00028 0.00358 0.00352 1 7 15 4 10 15 0.09103 0.18777
(0.4,0.6) 0.00042 0.03242 0.03240 2 6 8 2 7 10 0.67877 0.41021
(0.1,0.9) 0.00002 0.00090 0.00090 3 6 9 1 5 8 0.04456 0.04697
(0.8,0.2) 0.00015 0.01439 0.01440 1 6 10 3 8 13 0.00066 0.00484
Table 4.6 Simulations results (Product Copula).
126
In the previous results, it can be seen that in most cases the number of rejections are close to the
α-level. However in some cases we have large values in the columns that indicate the number
of rejections at α-level (rows in bold type). The problem occur in simulations where the point is
located near to the boundary of I2 = [0, 1] × [0, 1] and in simulations where the copula is close to
one of the Fréchet-Hoeffding bounds and the point is not in the support of the copula M or copula
W. This also happens if the copula is singular. As a possible solution in such cases, we propose to
increase the sample size or consider points very close to the support of the copula M or copula W to
obtain a better approximation to the normal distribution with the parameters indicated in Remark
4.27. For example, if we consider a sample size of 100,000 instead of a sample size of 50,000 in
the simulation of the Gumbel copula with θ = 20 in the point (0.8, 0.8) we obtain a mean of means
equal to −0.00039, a mean of variances equal to 0.00149, a real variance equal to 0.00153 and
4, 8 and 12 rejections of normality at α-level 0.01, 0.05 and 0.10, respectively, for the Anderson-
Darling’s test, and 1, 10 and 15 for the Shapiro-Wilk’s test, also we have a p-value of 0.79405 for
the Kolmogorov-Smirnov’s uniformity test applied to the p-values of the Anderson-Darling’s tests
and a p-value of 0.77178 for the Kolmogorov-Smirnov’s uniformity test applied to the p-values of
the Shapiro-Wilk’s tests.
127
Conclusions
We know that Glivenko-Cantelli’s Theorem 2.5, is one out of two results which has made the study
of the empirical distribution functions a really important task. In our Theorem 2.17 we proved
that the sample copula also satisfies a Glivenko-Cantelli’s result. It is unquestionable that the
empirical distribution function in dimension d = 1 is the best possible approximation to the real
distribution function, a proof of that is the Dvoretzky-Kiefer-Wolfowitz’s inequality, see [11] and
[34]. However, for dimension higher than one, even if Glivenko-Cantelli’s Theorem still holds, the
price we have to pay is that we need very large sample sizes in order to obtain a good approximation
of the real distribution function, this needed sample size increases dramatically with the dimension
d. Besides, the evaluation of the empirical distribution function needs arrays of order nd where
n is the sample size and this becomes a problem for large sample sizes even in relatively small
dimensions. On the other hand, for the sample d-copula or order m, if m is relatively small compare
to n the arrays we need for the construction of Cn
m is only of order md, which in general is a lot
smaller that nd.
We saw in every simulation of Chapter 2, that there exists a value of m for which Cn
m is a better
approximation than the empirical copula Cn. If we define m0 the value of 2 ≤ m ≤ n such that Cn
m0
is a better approximation Cn and it minimizes the mean value of the supremum distance between
Cn
m and C the real copula, then we would have had selected the best possible m. Of course, when
we have only one sample from an unknown distribution it is not easy to select an appropriate
value of m. In Section 5 of the Chapter 2, for dimension d = 2 we are proposing a method to
estimate the value of m based on the sample Spearman’s ρ. This idea is based on the fact that
the best approximation of Cn
m for Πd the product d-copula is always reached when m = 2 for any
sample size n, that is, when ρ = 0. This statement also suggests that we may use the total variation
between the sample copula and the product copula using only the densities induced by Πd and by
Cn
m in order to estimate the parameter m.
As a last interesting remark, let us assume that d = 1 and that we have a random sample X1, X2, . . . , Xn
from a continuous distribution F, let Yi = i/n for i ∈ In be the modified sample, and take 2 ≤ m ≤ n.
Then if we define the uniform partition of I = [0, 1] of size m and we construct Cn
m following the
same ideas as in Definition 2.12 for d ≥ 2, but in dimension one. We have that Cn
m(u) = u for every
u ∈ I. Therefore, Cn
m(u) is the distribution of F(X) independently of the selection of m. This state-
ment clearly indicates that using the ideas of Deheuvels [8] we always find the right distribution
independently of the real continuous F, the sample size n and the selected m.
The aim of Chapter 3 was to show that in addition to the comparative advantages shown in Chapter
128
2 between the sample d-copula of order m and the empirical copula, we can obtain similar results
to those existing for the empirical copula given in [8] and that these results can be extended to
higher dimensions, in this way we obtain a complete description of the distributions associated to
the sample copula under the independence assumption.
As we mentioned before, the second result that makes important the use of Fn, the empirical
distribution, or the empirical copula Cn is the central limit theorem. In the Chapter 4 we proved
the convergence to a Gaussian process of the sample copula process
√
n(Cn
m − C), from this result
we obtained a Central Limit Theorem similar to the existing one for the empirical copula process.
From the last two chapters of this thesis we can replicate, for the sample copula, the most important
results of the convergence of the empirical copula, and along with the results of Chapter 2 we can
consider that the sample copula could be a good approximation to the checkerboard copula and
therefore to the real copula in comparison to the approximation given for the empirical copula.
Among possible applications of the sample copula we have the study of the credit risk of the
financial institutions, in [32] a methodology is proposed to calculate the default correlations using
the Gaussian copula, although this work suffered several criticism because these models did not
provide adequate results under the conditions of the financial crisis of 2008, many efforts have
been made by using Gaussian model variants, see [3]. The sample copula can be used to model
the overall risk without assuming any specific family of copulas. We have also worked on an
algorithm to generate, from the sample copula, simulated samples that preserve the dependence
structure which may be used in methodologies related to tests of goodness of fit or parameter
estimation.
129
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