ISRN Probability and StatisticsVolume 2012 (2012), Article ID 468758, 7 pagesdoi:10.5402/2012/468758

Research Article

A Functional for Copulas and Quasi-Copulas

Manuel Úbeda-Flores

Department of Statistics and Applied Mathematics, University of Almería, Carretera de Sacramento s/n, La Cañada de San Urbano, 04120 Almería, Spain

Received 27 February 2012; Accepted 19 March 2012

Academic Editors: C. Proppe and C.-W. Wu

Copyright © 2012 Manuel Úbeda-Flores. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We recall and study some properties of a known functional operating on the set of -copulas and determine conditions under such functional is well defined on the set of -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 of a random vector —where is a natural number such that —with respective univariate margins , can be expressed in the form , , in terms of an -copula that is uniquely determined on Range .

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 -dimensional quasi-copula (or -quasi-copula) as a function that satisfies the following:(Q1) and for all and for every ;(Q2) is nondecreasing in each variable;(Q3)the 1-Lipschitz condition, that is, for each , then .

While every -copula is an -quasi-copula, there exist -quasi-copulas which are not -copulas; in this case it is said that is a proper -quasi-copula. Every -quasi-copula (and hence any -copula) satisfies the following inequalities: for each in . is an -copula for every ; and is a 2-copula, but is a proper -quasi-copula for every .

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 be a function defined on , and let denote the -box in such that for all . The function is said to be -increasing if , where the sum is taken over all the vertices of —that is, each is equal to either or —and is 1 if for an even number of s, and −1 if for an odd number of s. Thus, an -copula is an -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 -copulas using a known functional operating defined in [13]. Moreover, we also determine conditions under such functional is an -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 defined in by .

Let be a fixed -copula. Consider the functional defined for any -copula and any functions on into as follows: for every in with . Conditions under this function is an -copula for any -copula are given in the following result.

Theorem 2.1 ([13, Theorem 3.4]). Let be the function defined as (2.1). If (i)the function is -increasing; (ii) for every ;(iii)for every , there is some , , such that , and either one of the two following conditions holds;(iv)every function depends solely on one variable, which is different for each ; all ’s are monotone but the number of them that are decreasing is even;(v)there exists one variable such that none of the functions depends on . Then is an -copula for every -copula .

2.1. Examples

In [13], there are no examples of -copulas of type (2.1) satisfying the conditions in Theorem 2.1. In what follows, we provide examples of such -copulas, but first we need the following result, which shows that the convex linear combination of two -copulas is an -copula, and whose proof is immediate.

Proposition 2.2. Let and be two -copulas. Then, the function for all in with , is an -copula.

Thus, taking as any fixed -copula, the zero function—that is, for all —, and , for all in and for every , 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 is not identically equal to zero.

We now provide two examples.

Example 2.3. Let be a fixed -copula, , , and for all in . If is the function defined by , then for any -box , we have that is -increasing, since . 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 represent a family of -copulas for any -copula .

Example 2.4. Let be a natural number such that . Let for , and , , for every in . If is any -copula, we have that conditions (i), (ii), (iii), and (iv)—but not (v)—in Theorem 2.1 are satisfied. Thus, the functions given by represent a family of -copulas for any -copula .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 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 be a fixed -copula, and let , with , and , , for all in . 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 -copula , the function given by (2.1), that is, for every in , is an -copula, since the function is a member of the known Farlie-Gumbel-Morgenstern family of -copulas [14, equation (44.73)], and we only need to apply Proposition 2.2. For , we can choose any family of -copulas different from (see [2, 15–17] e.g).

2.2. Association and Dependence

For statistical modelling [18], with each -copula 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 (see [19, 20]), where denotes the survival function of [2].

A coefficient that summarize some statistical properties of a copula is introduced in [21]. Let be a random vector with joint distribution function and univariate margins . Let and . The lower extremal dependence coefficient of is defined as , and the upper extremal dependence coefficient of is defined as (if the limits exist). We obtain that, for every , in terms of the associated copula , these coefficients are given by respectively.

Since the general computation of these coefficients for the -copulas given by (2.1) do not give us much information, let us take the family of -copulas given by (2.4) with . Then, it is a simple exercise to show that and

2.3. Quasi-Copulas

Assume is a fixed -quasi-copula and consider the functional defined for any -quasi-copula and any functions on into as follows: for every in with . We want to study conditions for the functions which assure that is an -quasi-copula for any -quasi-copula —of course, Theorem 2.1 is valid for this case too. We have the following result.

Theorem 2.6. Let be the function defined by (2.9) for which properties (ii) and (iii) in Theorem 2.1 are satisfied. Moreover, suppose (vi)the functions and , , are increasing in each variable;(vii) satisfies the -Lipschitz condition with and satisfies the -Lipschitz condition for all . Then is an -quasi-copula for every -quasi-copula .

Proof. Condition (Q1) in the definition of -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 and be in such that . Since and and are -quasi-copulas—that is, and satisfy conditions (Q2) and (Q3); using condition (vi), we obtain immediately that , that is, satisfies (Q2); and using condition (vii) we have that Thus, for every in , we have that that is, 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 as any fixed -quasi-copula, an -quasi-copula, and the zero function for every . Thus, we have a similar result to Proposition 2.2 applied to -quasi-copulas.

Proposition 2.7. Let and be two -quasi-copulas. Then, the function for all in with , is an -quasi-copula.

Conditions (vi) and (vii) in Theorem 2.6 may be not necessary. For instance, if is the zero function, for every , then condition (vii) in Theorem 2.6 is not satisfied; however, in this case, we obtain a family of (maybe, proper) -quasi-copulas of type (2.9). Another example with not identically zero is the following.

Example 2.8. Let be the product -copula, and , , for every with . 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 is an -quasi-copula, since is a proper -quasi-copula for (note that the case corresponds to the product -copula). If, for instance, we take in (2.13), we obtain a family of proper -quasi-copulas.

3. Conclusion

In this note, we have recalled a known functional operating on the set of -copulas, provided examples of -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 -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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