# A Functional for Copulas and Quasi-Copulas

Authors: Manuel Úbeda-Flores
Journal: ISRN Probability and Statistics
 (2012)

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## Abstract

We recall and study some properties of a known functional operating on the set ofn-copulas and determine conditions under such functional is well defined on the set of n-quasi-copulas. As
a consequence, new families of copulas and quasi-copulas are defined, illustrating our results with several
examples.


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## Body

## 1. Introduction

The termcopula, coined by Sklar [1], is now common in the statistical literature (for a complete survey, see [2]). The importance of copulas as a tool for statistical analysis and modeling stems largely from the observation that the joint distribution function H of a random vector (X1,X2,…,Xn)—where n is a natural number such that n≥2—with respective univariate margins F1,F2,…,Fn, can be expressed in the form H(x)=C(F1(x1),F2(x2),…,Fn(xn)), x=(x1,x2,…,xn)∈[-∞,∞]n, in terms of an n-copula C that is uniquely determined on ×i=1n Range Fi.Copulas are becoming popular in development of quantitative risk management methodology within finance and insurance [3].Alsina et al. [4] introduced the notion of (bivariate) quasi-copula in order to characterize operations on distribution functions that can, or cannot, be derived from operations on random variables defined on the same probability space (for the multivariate case, see [5]). Cuculescu and Theodorescu [6] have characterized an n-dimensional quasi-copula (or n-quasi-copula) as a function Q:[0,1]n→[0,1] that satisfies the following:(Q1)
Q(u1,…,ui-1,0,ui+1,…,un)=0 and Q(1,…,1,ui,1,…,1)=ui for all u∈[0,1]n and for every i=1,2,…,n;(Q2)
Q is nondecreasing in each variable;(Q3)
the 1-Lipschitz condition, that is, for each u,v∈[0,1]n, then |Q(u)-Q(v)|≤∑i=1n|ui-vi|.While everyn-copula is an n-quasi-copula, there exist n-quasi-copulas Q which are not n-copulas; in this case it is said that Q is a propern-quasi-copula. Every n-quasi-copula Q (and hence any n-copula) satisfies the following inequalities: Wn(u)=max(∑i=1nui-n+1,0)≤Q(u)≤min(u1,u2,…,un)=Mn(u) for each u in [0,1]n. Mn is an n-copula for every n≥2; and W2 is a 2-copula, but Wn is a proper n-quasi-copula for every n≥3.In the last years an increasing interest has been devoted to these functions by researchers in some topics of fuzzy sets theory, such as preference modeling, similarities, and fuzzy logics (see [7] for an overview).Letf be a function defined on [0,1]n, and let B denote the n-boxB=×i=1n[ai,bi] in [0,1]n such that ai≤bi for all i=1,2,…,n. The function f is said to be n-increasing if ∑sgn(c)·f(c)≥0, where the sum is taken over all the verticesc=(c1,c2,…,cn) of B—that is, each ck is equal to either ak or bk—and sgn(c) is 1 if ck=ak for an even number of k′s, and −1 if ck=ak for an odd number of k′s. Thus, an n-copula is an n-increasing function satisfying condition (Q1). Some differences between copulas and quasi-copulas can be found in [8–12].In this note, we provide new families ofn-copulas using a known functional operating defined in [13]. Moreover, we also determine conditions under such functional is an n-quasi-copula. As a consequence, new families of copulas and quasi-copulas are defined.


## 2. The Functional

In the following, we will consider the setJ defined in [0,1]n by J={u∈[0,1]n∣u1u2…un=uifor somei,1≤i≤n}.LetP be a fixed n-copula. Consider the functional defined for any n-copula C and any functions f,α1,α2,…,αn on [0,1]n into [0,1] as follows:(2.1)C*(u)=λ⋅P(u)+(1-λ)⋅[f(u)+C(α1(u),α2(u),…,αn(u))]
for every u in [0,1]n with λ∈[0,1]. Conditions under this function is an n-copula for any n-copula C are given in the following result.Theorem 2.1 ([13, Theorem 3.4]).
LetC* be the function defined as (2.1). If
(i)
the functionf is n-increasing;(ii)
f(u)=u1u2…un-∏i=1nαi(u) for every u∈J;(iii)
for everyu∈J, there is some j, 1≤j≤n, such that ∏i=1nαi(u)=αj(u), and either one of the two following conditions holds;(iv)
every functionαi depends solely on one variable, which is different for each αi; all αi’s are monotone but the number of them that are decreasing is even;(v)
there exists one variableuj(1≤j≤n) such that none of the functions αi depends on uj.
ThenC* is an n-copula for every n-copula C.


### 2.1. Examples

In [13], there are no examples of n-copulas of type (2.1) satisfying the conditions in Theorem 2.1. In what follows, we provide examples of such n-copulas, but first we need the following result, which shows that the convex linear combination of two n-copulas is an n-copula, and whose proof is immediate.Proposition 2.2.
LetC1 and C2 be two n-copulas. Then, the function C(u)=θ·C1(u)+(1-θ)·C2(u) for all u in [0,1]n with θ∈[0,1],  is an n-copula.Thus, takingP as any fixed n-copula, f the zero function—that is, f(u)=0 for all u∈[0,1]n—, and αi(u)=ui, for all u in [0,1]n and for every i=1,2,…,n, in (2.1), then conditions (i)–(iv) in Theorem 2.1 are immediately satisfied, and hence we obtain the result in Proposition 2.2. In what follows we will find examples in which the function f is not identically equal to zero.We now provide two examples.Example 2.3.
LetP be a fixed n-copula,αi(u)=ui(1-ui),i=1,2,…,n-1,and αn(u)=1 for all u in [0,1]n.If f is the function defined by f(u)=u12u2…un,then for any n-box B=×i=1n[ui,vi],we have that f is n-increasing,since (v2+u2)∏i=1n(vi-ui)≥0.It is easy to check that conditions (i), (ii), (iii), and (v)—but not (iv)—in Theorem 2.1 are satisfied. Thus, we have that the functions given by
(2.2)C*(u)=λ⋅P(u)+(1-λ)⋅[u12u2…un+C(u1(1-u1),…,un-1(1-un-1),1)],u∈[0,1]n,
represent a family of n-copulas for any n-copula C.Example 2.4.
Letn be a natural number such that n≥3. Letαi(u)=1-uifori=1,2,3,and αi(u)=ui,i≥4,for every u in [0,1]n.If f is any n-copula,we have that conditions (i), (ii), (iii), and (iv)—but not (v)—in Theorem 2.1 are satisfied. Thus, the functions given by
(2.3)C*(u)=λ⋅P(u)+(1-λ)⋅[f(u)+C(1-u1,1-u2,1-u3,u4,…,un)],u∈[0,1]n,
represent a family of n-copulas for any n-copula C.
In [13], it is noted that conditions (iv) or (v) in Theorem 2.1 could be too strong, and the authors provide a counterexample in which none of the conditions is satisfied and the function C* given by (2.1) is not a copula. However, we stress that these two conditions are sufficient (but not necessary), as the following example shows.Example 2.5.
LetP be a fixed n-copula,and let f(u)=∏i=1nui,α1(u)=δ∏i=1nui(1-ui) with δ∈[-1,1], and αi(u)=1, i=2,…,n, for all u in [0,1]n.Then, it is easy to check that these functions satisfy conditions (i), (ii), and (iii) in Theorem 2.1, but neither (iv) nor (v) hold. However, for any n-copula C, the function given by (2.1), that is,(2.4)C*(u)=λ⋅P(u)+(1-λ)⋅[∏i=1nui+C(δ∏i=1nui(1-ui),1,…,1)]=λ⋅P(u)+(1-λ)⋅[∏i=1nui⋅(1+δ∏i=1n(1-ui))]
for everyu in [0,1]n, is an n-copula, since the function D(u)=∏i=1nui·[1+δ∏i=1n(1-ui)] is a member of the known Farlie-Gumbel-Morgenstern family of n-copulas [14, equation (44.73)], and we only need to apply Proposition 2.2. For P, we can choose any family of n-copulas different from D (see [2, 15–17] e.g).


### 2.2. Association and Dependence

For statistical modelling [18], with each n-copula C we can associate, among others, a non-parametric measure of multivariate association, called the medial correlation coefficient (or Blomqvist’s beta), which can be easily computed as
(2.5)βn(C)=2n-1[C(1/2)+C¯(1/2)]-12n-1-1
(see [19, 20]), where C¯ denotes the survival function of C [2].A coefficient that summarize some statistical properties of a copula is introduced in [21]. Let X be a random vector with joint distribution function H and univariate margins F1,…,Fn. Let Fmin:=min(F1(X1),…,Fn(Xn)) and Fmax:=max(F1(X1),…,Fn(Xn)). The lower extremal dependence coefficient of X is defined as εL:=limt→0+P[Fmax≤t∣Fmin≤t], and the upper extremal dependence coefficient of X is defined as εU:=limt→1-P[Fmin>t∣Fmax>t] (if the limits exist). We obtain that, for every t=(t,…,t)∈[0,1]n, in terms of the associated copula C, these coefficients are given by
(2.6)εL=lim⁡t→0+C(t)1-C¯(t)εU=lim⁡t→1-C¯(t)1-C(t),
respectively.Since the general computation of these coefficients for then-copulas given by (2.1) do not give us much information, let us take the family of n-copulas given by (2.4) with P=Mn. Then, it is a simple exercise to show that
(2.7)βn(C*)={λ+δ(1-λ)22n-1,ifnisevenλ,ifnisodd,
and
(2.8)εL=εU=λλ+(1-λ)n.


### 2.3. Quasi-Copulas

AssumeP is a fixed n-quasi-copula and consider the functional defined for any n-quasi-copula Q and any functions f,α1,α2,…,αn on [0,1]n into [0,1] as follows:(2.9)Q*(u)=λ⋅P(u)+(1-λ)⋅[f(u)+Q(α1(u),α2(u),…,αn(u))]
for every u in [0,1]n with λ∈[0,1]. We want to study conditions for the functions f,α1,α2,…,αn which assure that Q* is an n-quasi-copula for any n-quasi-copula Q—of course, Theorem 2.1 is valid for this case too. We have the following result.Theorem 2.6.
LetQ* be the function defined by (2.9) for which properties (ii) and (iii) in Theorem 2.1 are satisfied. Moreover, suppose
(vi)
the functionsf and αi, i=1,2,…,n, are increasing in each variable;(vii)
f satisfies the r-Lipschitz condition with r∈[0,1] and αi satisfies the (1-r)/n-Lipschitz condition for all i=1,2,…,n.
ThenQ* is an n-quasi-copula for every n-quasi-copula Q.Proof.
Condition (Q1) in the definition ofn-quasi-copula is equivalent to the conditions (ii) and (iii) in Theorem 2.1 [13]. We prove now that conditions (Q2) and (Q3) are satisfied. For that, let u′=(u1,…,ui-1,ui′,ui+1,…,un) and u=(u1,…,ui,…,un) be in [0,1]n such that ui≤ui′. Since
(2.10)Q*(u′)-Q*(u)=λ⋅P(u′)+(1-λ)⋅[f(u′)+Q(α1(u′),…,αn(u′))]-λ⋅P(u)-(1-λ)⋅[f(u)+Q(α1(u),…,αn(u))]=λ⋅[P(u′)-P(u)]+(1-λ)⋅[f(u′)-f(u)+Q(α1(u′),…,αn(u′))-Q(α1(u),…,αn(u))],and P and Q are n-quasi-copulas—that is, P and Q satisfy conditions (Q2) and (Q3); using condition (vi), we obtain immediately that Q*(u′)-Q*(u)≥0, that is, Q* satisfies (Q2); and using condition (vii) we have that
(2.11)Q*(u′)-Q*(u)≤λ⋅[P(u′)-P(u)]+(1-λ)⋅(f(u′)-f(u)+∑i=1n[αi(u′)-αi(u)])≤λ⋅(ui′-ui)+(1-λ)⋅[r(ui′-ui)+n⋅1-rn⋅(ui′-ui)]=(ui′-ui).
Thus, for every u,v in [0,1]n, we have that
(2.12)|Q*(v)-Q*(u)|=|Q*(v1,…,vn)-Q*(v1,…,vn-1,un)|+|Q*(v1,…,vn-1,un)-Q*(v1,…,vn-2,vn-1,un)|+⋯+|Q*(v1,u2,…,un)-Q*(u1,…,un)|≤∑i=1n|vi-ui|,
that is, Q* satisfies (Q3), which completes the proof.As an application of Theorem2.6, we can generalize the result in Proposition 2.2 in the following sense: take P as any fixed n-quasi-copula, f an n-quasi-copula, and αi the zero function for every i=1,2,…,n. Thus, we have a similar result to Proposition 2.2 applied to n-quasi-copulas.Proposition 2.7.
LetQ1 and  Q2 be two n-quasi-copulas.  Then, the function Q(u)=θ·Q1(u)+(1-θ)·Q2(u) for all u in [0,1]n with θ∈[0,1],  is an n-quasi-copula.Conditions (vi) and (vii) in Theorem2.6 may be not necessary. For instance, if f is the zero function, αi(u)=uj for every i,j=1,2,…,n, then condition (vii) in Theorem 2.6 is not satisfied; however, in this case, we obtain a family of (maybe, proper) n-quasi-copulas of type (2.9). Another example with f not identically zero is the following.Example 2.8.
Letf be the product n-copula,α1(u)=γ(1-un)(1-max1≤i≤n-1ui)∏i=1nui and αi(u)=1,i=2,…,n,for every u∈[0,1]n with γ∈[0,1].Then, conditions (ii) and (iii) in Theorem 2.1 hold, but condition (vi) in Theorem 2.6 is not satisfied. However, via Proposition 2.7, we have that
(2.13)Q*(u)=λP(u)+(1-λ)⋅∏i=1nui[1+γ(1-un)(1-max⁡1≤i≤n-1ui)]
is an n-quasi-copula, since ∏i=1nui[1+γ(1-un)(1-max1≤i≤n-1ui)] is a proper n-quasi-copula for γ∈(0,1] (note that the case γ=0 corresponds to the product n-copula). If, for instance, we take P=Wn in (2.13), we obtain a family of proper n-quasi-copulas.


## 2.1. Examples

In [13], there are no examples of n-copulas of type (2.1) satisfying the conditions in Theorem 2.1. In what follows, we provide examples of such n-copulas, but first we need the following result, which shows that the convex linear combination of two n-copulas is an n-copula, and whose proof is immediate.Proposition 2.2.
LetC1 and C2 be two n-copulas. Then, the function C(u)=θ·C1(u)+(1-θ)·C2(u) for all u in [0,1]n with θ∈[0,1],  is an n-copula.Thus, takingP as any fixed n-copula, f the zero function—that is, f(u)=0 for all u∈[0,1]n—, and αi(u)=ui, for all u in [0,1]n and for every i=1,2,…,n, in (2.1), then conditions (i)–(iv) in Theorem 2.1 are immediately satisfied, and hence we obtain the result in Proposition 2.2. In what follows we will find examples in which the function f is not identically equal to zero.We now provide two examples.Example 2.3.
LetP be a fixed n-copula,αi(u)=ui(1-ui),i=1,2,…,n-1,and αn(u)=1 for all u in [0,1]n.If f is the function defined by f(u)=u12u2…un,then for any n-box B=×i=1n[ui,vi],we have that f is n-increasing,since (v2+u2)∏i=1n(vi-ui)≥0.It is easy to check that conditions (i), (ii), (iii), and (v)—but not (iv)—in Theorem 2.1 are satisfied. Thus, we have that the functions given by
(2.2)C*(u)=λ⋅P(u)+(1-λ)⋅[u12u2…un+C(u1(1-u1),…,un-1(1-un-1),1)],u∈[0,1]n,
represent a family of n-copulas for any n-copula C.Example 2.4.
Letn be a natural number such that n≥3. Letαi(u)=1-uifori=1,2,3,and αi(u)=ui,i≥4,for every u in [0,1]n.If f is any n-copula,we have that conditions (i), (ii), (iii), and (iv)—but not (v)—in Theorem 2.1 are satisfied. Thus, the functions given by
(2.3)C*(u)=λ⋅P(u)+(1-λ)⋅[f(u)+C(1-u1,1-u2,1-u3,u4,…,un)],u∈[0,1]n,
represent a family of n-copulas for any n-copula C.
In [13], it is noted that conditions (iv) or (v) in Theorem 2.1 could be too strong, and the authors provide a counterexample in which none of the conditions is satisfied and the function C* given by (2.1) is not a copula. However, we stress that these two conditions are sufficient (but not necessary), as the following example shows.Example 2.5.
LetP be a fixed n-copula,and let f(u)=∏i=1nui,α1(u)=δ∏i=1nui(1-ui) with δ∈[-1,1], and αi(u)=1, i=2,…,n, for all u in [0,1]n.Then, it is easy to check that these functions satisfy conditions (i), (ii), and (iii) in Theorem 2.1, but neither (iv) nor (v) hold. However, for any n-copula C, the function given by (2.1), that is,(2.4)C*(u)=λ⋅P(u)+(1-λ)⋅[∏i=1nui+C(δ∏i=1nui(1-ui),1,…,1)]=λ⋅P(u)+(1-λ)⋅[∏i=1nui⋅(1+δ∏i=1n(1-ui))]
for everyu in [0,1]n, is an n-copula, since the function D(u)=∏i=1nui·[1+δ∏i=1n(1-ui)] is a member of the known Farlie-Gumbel-Morgenstern family of n-copulas [14, equation (44.73)], and we only need to apply Proposition 2.2. For P, we can choose any family of n-copulas different from D (see [2, 15–17] e.g).


## 2.2. Association and Dependence

For statistical modelling [18], with each n-copula C we can associate, among others, a non-parametric measure of multivariate association, called the medial correlation coefficient (or Blomqvist’s beta), which can be easily computed as
(2.5)βn(C)=2n-1[C(1/2)+C¯(1/2)]-12n-1-1
(see [19, 20]), where C¯ denotes the survival function of C [2].A coefficient that summarize some statistical properties of a copula is introduced in [21]. Let X be a random vector with joint distribution function H and univariate margins F1,…,Fn. Let Fmin:=min(F1(X1),…,Fn(Xn)) and Fmax:=max(F1(X1),…,Fn(Xn)). The lower extremal dependence coefficient of X is defined as εL:=limt→0+P[Fmax≤t∣Fmin≤t], and the upper extremal dependence coefficient of X is defined as εU:=limt→1-P[Fmin>t∣Fmax>t] (if the limits exist). We obtain that, for every t=(t,…,t)∈[0,1]n, in terms of the associated copula C, these coefficients are given by
(2.6)εL=lim⁡t→0+C(t)1-C¯(t)εU=lim⁡t→1-C¯(t)1-C(t),
respectively.Since the general computation of these coefficients for then-copulas given by (2.1) do not give us much information, let us take the family of n-copulas given by (2.4) with P=Mn. Then, it is a simple exercise to show that
(2.7)βn(C*)={λ+δ(1-λ)22n-1,ifnisevenλ,ifnisodd,
and
(2.8)εL=εU=λλ+(1-λ)n.


## 2.3. Quasi-Copulas

AssumeP is a fixed n-quasi-copula and consider the functional defined for any n-quasi-copula Q and any functions f,α1,α2,…,αn on [0,1]n into [0,1] as follows:(2.9)Q*(u)=λ⋅P(u)+(1-λ)⋅[f(u)+Q(α1(u),α2(u),…,αn(u))]
for every u in [0,1]n with λ∈[0,1]. We want to study conditions for the functions f,α1,α2,…,αn which assure that Q* is an n-quasi-copula for any n-quasi-copula Q—of course, Theorem 2.1 is valid for this case too. We have the following result.Theorem 2.6.
LetQ* be the function defined by (2.9) for which properties (ii) and (iii) in Theorem 2.1 are satisfied. Moreover, suppose
(vi)
the functionsf and αi, i=1,2,…,n, are increasing in each variable;(vii)
f satisfies the r-Lipschitz condition with r∈[0,1] and αi satisfies the (1-r)/n-Lipschitz condition for all i=1,2,…,n.
ThenQ* is an n-quasi-copula for every n-quasi-copula Q.Proof.
Condition (Q1) in the definition ofn-quasi-copula is equivalent to the conditions (ii) and (iii) in Theorem 2.1 [13]. We prove now that conditions (Q2) and (Q3) are satisfied. For that, let u′=(u1,…,ui-1,ui′,ui+1,…,un) and u=(u1,…,ui,…,un) be in [0,1]n such that ui≤ui′. Since
(2.10)Q*(u′)-Q*(u)=λ⋅P(u′)+(1-λ)⋅[f(u′)+Q(α1(u′),…,αn(u′))]-λ⋅P(u)-(1-λ)⋅[f(u)+Q(α1(u),…,αn(u))]=λ⋅[P(u′)-P(u)]+(1-λ)⋅[f(u′)-f(u)+Q(α1(u′),…,αn(u′))-Q(α1(u),…,αn(u))],and P and Q are n-quasi-copulas—that is, P and Q satisfy conditions (Q2) and (Q3); using condition (vi), we obtain immediately that Q*(u′)-Q*(u)≥0, that is, Q* satisfies (Q2); and using condition (vii) we have that
(2.11)Q*(u′)-Q*(u)≤λ⋅[P(u′)-P(u)]+(1-λ)⋅(f(u′)-f(u)+∑i=1n[αi(u′)-αi(u)])≤λ⋅(ui′-ui)+(1-λ)⋅[r(ui′-ui)+n⋅1-rn⋅(ui′-ui)]=(ui′-ui).
Thus, for every u,v in [0,1]n, we have that
(2.12)|Q*(v)-Q*(u)|=|Q*(v1,…,vn)-Q*(v1,…,vn-1,un)|+|Q*(v1,…,vn-1,un)-Q*(v1,…,vn-2,vn-1,un)|+⋯+|Q*(v1,u2,…,un)-Q*(u1,…,un)|≤∑i=1n|vi-ui|,
that is, Q* satisfies (Q3), which completes the proof.As an application of Theorem2.6, we can generalize the result in Proposition 2.2 in the following sense: take P as any fixed n-quasi-copula, f an n-quasi-copula, and αi the zero function for every i=1,2,…,n. Thus, we have a similar result to Proposition 2.2 applied to n-quasi-copulas.Proposition 2.7.
LetQ1 and  Q2 be two n-quasi-copulas.  Then, the function Q(u)=θ·Q1(u)+(1-θ)·Q2(u) for all u in [0,1]n with θ∈[0,1],  is an n-quasi-copula.Conditions (vi) and (vii) in Theorem2.6 may be not necessary. For instance, if f is the zero function, αi(u)=uj for every i,j=1,2,…,n, then condition (vii) in Theorem 2.6 is not satisfied; however, in this case, we obtain a family of (maybe, proper) n-quasi-copulas of type (2.9). Another example with f not identically zero is the following.Example 2.8.
Letf be the product n-copula,α1(u)=γ(1-un)(1-max1≤i≤n-1ui)∏i=1nui and αi(u)=1,i=2,…,n,for every u∈[0,1]n with γ∈[0,1].Then, conditions (ii) and (iii) in Theorem 2.1 hold, but condition (vi) in Theorem 2.6 is not satisfied. However, via Proposition 2.7, we have that
(2.13)Q*(u)=λP(u)+(1-λ)⋅∏i=1nui[1+γ(1-un)(1-max⁡1≤i≤n-1ui)]
is an n-quasi-copula, since ∏i=1nui[1+γ(1-un)(1-max1≤i≤n-1ui)] is a proper n-quasi-copula for γ∈(0,1] (note that the case γ=0 corresponds to the product n-copula). If, for instance, we take P=Wn in (2.13), we obtain a family of proper n-quasi-copulas.


## 3. Conclusion

In this note, we have recalled a known functional operating on the set ofn-copulas, provided examples of n-copulas satisfying the conditions in Theorem 2.1, and studied some properties of association and dependence. Finally, we have determined conditions under such functional are well defined on the set of n-quasi-copulas.



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