A Functional for Copulas and Quasi-Copulas

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 ∈ −∞,∞ , in terms of an n-copula C that is uniquely determined on ×i 1 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 nquasi-copula as a function Q : 0, 1 n → 0, 1 that satisfies the following:


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 X 1 , X 2 , . . ., X n -where n is a natural number such that n ≥ 2with respective univariate margins F 1 , F 2 , . . ., F n , can be expressed in the form H x C F 1 x 1 , F 2 x 2 , . . ., F n x n , x x 1 , x 2 , . . ., x n ∈ −∞, ∞ n , in terms of an n-copula C that is uniquely determined on × n i 1 Range F i .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 nquasi-copula as a function Q : 0, 1 n → 0, 1 that satisfies the following: u i for all u ∈ 0, 1 n and for every i 1, 2, . . ., n; 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: M n u for each u in 0, 1 n .M n is an n-copula for every n ≥ 2; and W 2 is a 2-copula, but W n 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 × n i 1 a i , b i in 0, 1 n such that a i ≤ b i 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 c 1 , c 2 , . . ., c n of B-that is, each c k is equal to either a k or b k -and sgn c is 1 if c k a k for an even number of k s, and −1 if c k a k 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-quasicopula.As a consequence, new families of copulas and quasi-copulas are defined.

The Functional
In the following, we will consider the set J defined in 0, 1 n by J {u ∈ 0, 1 n | u 1 u 2 . . .u n u i for some i, 1 ≤ i ≤ n}.
Let P be a fixed n-copula.Consider the functional defined for any n-copula C and any functions f, α 1 , α 2 , . . ., α n on 0, 1 n into 0, 1 as follows: 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.

and either one of the two following conditions holds;
iv every function α i depends solely on one variable, which is different for each α i ; all α i 's are monotone but the number of them that are decreasing is even; v there exists one variable u j 1 ≤ j ≤ n such that none of the functions α i depends on u j .
Then C * is an n-copula for every n-copula C.

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 ncopula, and whose proof is immediate.
Proposition 2.2.Let C 1 and C 2 be two n-copulas.Then, the function Thus, taking P as any fixed n-copula, f the zero function-that is, f u 0 for all u ∈ 0, 1 n -, and α i u u i , 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, α i u 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 n-copulas for any n-copula C.
Example 2.4.Let n be a natural number such that n ≥ 3. Let α i u 1 − u i for i 1, 2, 3, and α i u u i , 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 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 and α i u 1, i 2, . . ., n, for all u in 0, 1 n .Then, it is easy to check that these 4 ISRN Probability and Statistics 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, for every u in 0, 1 n , is an n-copula, since the function D 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 .

Association and Dependence
For statistical modelling 18 , with each n-copula C we can associate, among others, a nonparametric measure of multivariate association, called the medial correlation coefficient or Blomqvist's beta , which can be easily computed as 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 F 1 , . . ., F n .Let F min : min F 1 X 1 , . . ., F n X n and F max : max F 1 X 1 , . . ., F n X n .The lower extremal dependence coefficient of X is defined as ε L : lim t → 0 P F max ≤ t | F min ≤ t , and the upper extremal dependence coefficient of X is defined as ε U : lim t → 1 − P F min > t | F max > 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 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 M n .Then, it is a simple exercise to show that and

Quasi-Copulas
Assume P is a fixed n-quasi-copula and consider the functional defined for any n-quasicopula Q and any functions f, α 1 , α 2 , . . ., α n on 0, 1 n into 0, 1 as follows: for every u in 0, 1 n with λ ∈ 0, 1 .We want to study conditions for the functions f, α 1 , α 2 , . . ., α n 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 vi the functions f and α i , i 1, 2, . . ., n, are increasing in each variable; vii f satisfies the r-Lipschitz condition with r ∈ 0, 1 and α i 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 u 1 , . . ., u i−1 , u i , u i 1 , . . ., u n and u u 1 , . . ., u i , . . ., u n be in 0, 1 n such that u i ≤ u i .Since 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

ISRN Probability and Statistics
Thus, for every u, v in 0, 1 n , we have that 12 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 α i the zero function for every i 1, 2, . . ., n.Thus, we have a similar result to Proposition 2.2 applied to n-quasi-copulas.
Conditions vi and vii in Theorem 2.6 may be not necessary.For instance, if f is the zero function, α i u u j 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, α 1 u γ 1 − u n 1 − max 1≤i≤n−1 u i n i 1 u i and α i 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 13 is an n-quasi-copula, since n i 1 u i 1 γ 1 − u n 1 − max 1≤i≤n−1 u i 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 W n in 2.13 , we obtain a family of proper n-quasi-copulas.

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.

see 19 ,
20 , where C denotes the survival function of C 2 .