A Functional for Copulas and Quasi-Copulas

Abstract

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,…,𝑋_𝑛)—where 𝑛 is a natural number such that 𝑛≥2—with respective univariate margins 𝐹₁,𝐹₂,…,𝐹_𝑛, can be expressed in the form 𝐻(𝐱)=𝐶(𝐹₁(𝑥₁),𝐹₂(𝑥₂),…,𝐹_𝑛(𝑥_𝑛)), 𝐱=(𝑥₁,𝑥₂,…,𝑥_𝑛)∈[−∞,∞]^𝑛, in terms of an 𝑛-copula 𝐶 that is uniquely determined on ×^𝑛𝑖=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 𝑛-quasi-copula) as a function 𝑄∶[0,1]^𝑛→[0,1] that satisfies the following:(Q1)𝑄(𝑢₁,…,𝑢_𝑖−1,0,𝑢_𝑖+1,…,𝑢_𝑛)=0 and 𝑄(1,…,1,𝑢_𝑖,1,…,1)=𝑢_𝑖 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)|≤∑^𝑛𝑖=1|𝑢_𝑖−𝑣_𝑖|.

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: 𝑊^𝑛(𝐮)=max(∑^𝑛𝑖=1𝑢_𝑖−𝑛+1,0)≤𝑄(𝐮)≤min(𝑢₁,𝑢₂,…,𝑢_𝑛)=𝑀^𝑛(𝐮) for each 𝐮 in [0,1]^𝑛. 𝑀^𝑛 is an 𝑛-copula for every 𝑛≥2; and 𝑊² 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=×^𝑛𝑖=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=(𝑐₁,𝑐₂,…,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-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…𝑢_𝑛=𝑢_𝑖forsome𝑖,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,…,𝛼_𝑛 on [0,1]^𝑛 into [0,1] as follows:𝐶^∗(𝐮)=𝜆⋅𝑃(𝐮)+(1−𝜆)⋅𝑓(𝐮)+𝐶𝛼₁(𝐮),𝛼₂(𝐮),…,𝛼_𝑛(𝐮)(2.1) 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
(i)the function 𝑓 is 𝑛-increasing; (ii)𝑓(𝐮)=𝑢₁𝑢₂…𝑢_𝑛−∏^𝑛𝑖=1𝛼_𝑖(𝐮) for every 𝐮∈𝐽;(iii)for every 𝐮∈𝐽, there is some 𝑗, 1≤𝑗≤𝑛, such that ∏^𝑛𝑖=1𝛼_𝑖(𝐮)=𝛼_𝑗(𝐮), 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.

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 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)=𝑢_𝑖(1−𝑢_𝑖), 𝑖=1,2,…,𝑛−1, and 𝛼_𝑛(𝐮)=1 for all 𝐮 in [0,1]^𝑛. If 𝑓 is the function defined by 𝑓(𝐮)=𝑢²¹𝑢₂…𝑢_𝑛, then for any 𝑛-box 𝐵=×^𝑛𝑖=1[𝑢_𝑖,𝑣_𝑖], we have that 𝑓 is 𝑛-increasing, since (𝑣₂+𝑢₂)∏^𝑛𝑖=1(𝑣_𝑖−𝑢_𝑖)≥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)⋅𝑢²¹𝑢₂…𝑢_𝑛+𝐶𝑢₁1−𝑢₁,…,𝑢_𝑛−11−𝑢_𝑛−1,1,𝐮∈[0,1]^𝑛,(2.2) 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 𝐶^∗(𝐮)=𝜆⋅𝑃(𝐮)+(1−𝜆)⋅𝑓(𝐮)+𝐶1−𝑢₁,1−𝑢₂,1−𝑢₃,𝑢₄,…,𝑢_𝑛,𝐮∈[0,1]^𝑛,(2.3) 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)=∏^𝑛𝑖=1𝑢_𝑖, 𝛼₁(𝐮)=𝛿∏^𝑛𝑖=1𝑢_𝑖(1−𝑢_𝑖) with 𝛿∈[−1,1], and 𝛼_𝑖(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)⋅^𝑛_𝑖=1𝑢_𝑖+𝐶𝛿^𝑛_𝑖=1=𝜆⋅𝑃(𝐮)+(1−𝜆)⋅_𝑛𝑢_𝑖1−𝑢_𝑖,1,…,1_𝑖=1𝑢_𝑖⋅1+𝛿^𝑛_𝑖=11−𝑢_𝑖(2.4)
for every 𝐮 in [0,1]^𝑛, is an 𝑛-copula, since the function 𝐷(𝐮)=∏^𝑛𝑖=1𝑢_𝑖⋅[1+𝛿∏^𝑛𝑖=1(1−𝑢_𝑖)] 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 𝛽_𝑛(𝐶)=2^𝑛−1𝐶(𝟏/𝟐)+𝐶(𝟏/𝟐)−12^𝑛−1−1(2.5) (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,…,𝐹_𝑛. Let 𝐹_min∶=min(𝐹₁(𝑋₁),…,𝐹_𝑛(𝑋_𝑛)) and 𝐹_max∶=max(𝐹₁(𝑋₁),…,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 𝜀_𝐿=lim_𝑡→0+𝐶(𝐭)1−𝐶(𝐭)𝜀^𝑈=lim_𝑡→1−𝐶(𝐭)1−𝐶(𝐭),(2.6) 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 𝛽_𝑛𝐶^∗=𝜆+𝛿(21−^2𝑛−1𝜆)𝜆,,iiff𝑛𝑛iisseovdedn,(2.7) and 𝜀_𝐿=𝜀_𝑈=𝜆𝜆+(1−𝜆)𝑛.(2.8)

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,…,𝛼_𝑛 on [0,1]^𝑛 into [0,1] as follows:𝑄^∗(𝐮)=𝜆⋅𝑃(𝐮)+(1−𝜆)⋅𝑓(𝐮)+𝑄𝛼₁(𝐮),𝛼₂(𝐮),…,𝛼_𝑛(𝐮)(2.9) for every 𝐮 in [0,1]^𝑛 with 𝜆∈[0,1]. We want to study conditions for the functions 𝑓,𝛼₁,𝛼₂,…,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^=(𝑢₁,…,𝑢_𝑖−1,𝑢_𝑖,𝑢_𝑖+1,…,𝑢_𝑛) and 𝐮=(𝑢₁,…,𝑢_𝑖,…,𝑢_𝑛) be in [0,1]^𝑛 such that 𝑢_𝑖≤𝑢_𝑖. Since 𝑄^∗(𝐮′)−𝑄^∗(𝐮)=−𝜆𝜆⋅𝑃⋅𝑃𝐮(𝐮^)+−((11−−𝜆𝜆))⋅⋅𝑓𝑓(𝐮𝐮^)++𝑄𝑄𝛼𝛼₁₁𝐮^,…,𝛼_𝑛𝐮^=𝜆⋅𝑃𝐮_(𝐮),…,𝛼_𝑛(𝐮)⋅𝑓𝐮−𝑃(𝐮)+(1−𝜆)_−𝑓(𝐮)+𝑄𝛼₁𝐮^,…,𝛼_𝑛𝐮^−𝑄𝛼₁(𝐮),…,𝛼_𝑛(𝐮),(2.10)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)+^𝑛_𝑖=1𝛼_𝑖≤𝜆⋅𝑢_𝑖(𝐮′)−𝛼_𝑖(𝐮)−𝑢_𝑖+(1−𝜆)⋅𝑟𝑢_𝑖−𝑢_𝑖+𝑛⋅1−𝑟𝑛⋅𝑢^𝑖=𝑢_𝑖−𝑢_𝑖−𝑢_𝑖.(2.11) Thus, for every 𝐮,𝐯 in [0,1]^𝑛, we have that ||𝑄^∗(𝐯)−𝑄^∗(𝐮)||=+||𝑄||𝑄^∗_∗𝑣₁,…,𝑣_𝑛−𝑄^∗𝑣₁,…,𝑣_𝑛−1,𝑢_𝑛||+⋯+𝑣||𝑄₁,_∗…,𝑣_𝑛−1,𝑢_𝑛−𝑄^∗𝑣₁,…,𝑣_𝑛−2,𝑣_𝑛−1,𝑢_𝑛||≤_𝑛𝑣₁,𝑢₂,…,𝑢_𝑛−𝑄^∗𝑢₁,…,𝑢_𝑛||_𝑖=1||𝑣_𝑖−𝑢_𝑖||,(2.12) 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.

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, 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, 𝛼₁(𝐮)=𝛾(1−𝑢_𝑛)(1−max_{1≤𝑖≤𝑛−1}𝑢_𝑖)∏^𝑛𝑖=1𝑢_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 𝑄^∗(𝐮)=𝜆𝑃(𝐮)+(1−𝜆)⋅^𝑛_𝑖=1𝑢_𝑖1+𝛾1−𝑢^𝑛1−max_{1≤𝑖≤𝑛−1}𝑢_𝑖(2.13) is an 𝑛-quasi-copula, since ∏^𝑛𝑖=1𝑢_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.

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.