We recall and study some properties of a known functional operating on the set of n-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.
1. Introduction
The term copula, 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:
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;
Q is nondecreasing in each variable;
the 1-Lipschitz condition, that is, for each u,v∈[0,1]n, then |Q(u)-Q(v)|≤∑i=1n|ui-vi|.
While every n-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 proper n-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).
Let f be a function defined on [0,1]n, and let B denote the n-box B=×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 vertices c=(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 of n-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 set J defined in [0,1]n by J={u∈[0,1]n∣u1u2…un=uifor somei,1≤i≤n}.
Let P 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: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]).
Let C* be the function defined as (2.1). If
the function f is n-increasing;
f(u)=u1u2…un-∏i=1nαi(u) for every u∈J;
for every u∈J, there is some j, 1≤j≤n, such that ∏i=1nαi(u)=αj(u), and either one of the two following conditions holds;
every function αi 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;
there exists one variable uj(1≤j≤n) such that none of the functions αi depends on uj.
Then C* 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.
Let C1 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, taking P 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.
Let P 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
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.
Let n be a natural number such that n≥3. Let αi(u)=1-ui for i=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
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.
Let P 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,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 every u 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
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
εL=limt→0+C(t)1-C¯(t)εU=limt→1-C¯(t)1-C(t),
respectively.
Since the general computation of these coefficients for the n-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
βn(C*)={λ+δ(1-λ)22n-1,ifnisevenλ,ifnisodd,
and
εL=εU=λλ+(1-λ)n.
2.3. Quasi-Copulas
Assume P 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: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.
Let Q* be the function defined by (2.9) for which properties (ii) and (iii) in Theorem 2.1 are satisfied. Moreover, suppose
the functions f and αi, i=1,2,…,n, are increasing in each variable;
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.
Then Q* is an n-quasi-copula for every n-quasi-copula Q.
Proof.
Condition (Q1) in the definition of n-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
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
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
|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 Theorem 2.6, 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.
Let Q1 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 Theorem 2.6 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.
Let f 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
Q*(u)=λP(u)+(1-λ)⋅∏i=1nui[1+γ(1-un)(1-max1≤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 of n-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.
Acknowledgments
The author is grateful for the support by the Ministerio de Ciencia e Innovación (Spain) and FEDER, under Research Project MTM2009-08724.
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