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d-Dimensional dependence functions and Archimax copulas
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Fuzzy Sets and Systems 228 (2013) 78 – 87 www.elsevier.com/locate/fss
d-Dimensional dependence functions and Archimax copulas Radko Mesiar∗ Vladimír Jágr Slovak University of Technology in Bratislava, Slovakia Available online 2 November 2012
Abstract We introduce a new class of dependence functions and method for constructing tail and Pickands dependence functions. Finally, we conjecture the structure of d-dimensional Archimax copulas and discuss some classes of d-dimensional Archimax copulas. © 2012 Elsevier B.V. All rights reserved. Keywords: Copula; Extreme value copula; Archimedean copula; Archimax copula; Tail dependence function; Pickands dependence function
1. Introduction We suppose the reader to be familiar with the basic definitions and properties of copulas. For an overview for unfamiliar readers we recommend the monographs [12,15,16,27], a recent special issue of Information Sciences [8], and the recent papers [10,14]. While the case of two-dimensional copulas is discussed in many papers and books, including the proposal of several classes of two-dimensional copulas and construction methods for them, there are only few known classes of d-dimensional copulas with d > 2. We denote by Cd the set of all d-dimensional copulas. Recall that Cd is a convex and compact set (in the uniform convergence topology) of functions. Two distinguished classes in Cd are the extreme value copulas Ed (related to tail dependence functions, which when normed are called Pickands dependence functions), and the Archimedean copulas Ad (related to additive generators). For d = 2, there is an interesting extension of both C2 and A2 , namely the class AM2 of two-dimensional Archimax copulas. The aim of this paper is to discuss d-dimensional dependence functions and d-dimensional Archimax copulas, i.e., the class AMd , d > 2. Note that the preliminary version of this paper was presented at the summer school AGOP’2011 in Benevento, and here we have added several new results. The paper is organized as follows. In the next section, we recall extreme value copulas and d-dimensional Archimedean copulas. Note that this section is taken mostly from [13,24]. In Section 3, some well known d-dimensional dependence functions are recalled and a new class of such functions is introduced. It includes also a proposal of a construction method for tail dependence functions and Pickands dependence functions. In Section 4 we recall first the class AM2 of two-dimensional Archimax copulas [5]. Then we define the class AMd of d-dimensional Archimax copulas d > 2, and introduce some subclasses of AMd . Moreover, we introduce d-dimensional copulas based on a new class of d-dimensional dependence functions given in Section 3 and on additive generators of k-dimensional Archimedean copulas with k < d. Finally, some concluding remarks are given. ∗ Corresponding author.
E-mail addresses: [email protected] (R. Mesiar), [email protected] (V. Jágr). 0165-0114/$ - see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.fss.2012.10.010
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2. Extreme value copulas and Archimedean copulas This section is based on an overview paper on extreme-value copulas [13] and on [24] devoted to the study of d-dimensional Archimedean copulas. Extreme-value copulas not only arise naturally in the domain of extreme events, but they can also be a convenient choice to model data with positive dependence. An advantage with respect to the much more popular class of Archimedean copulas, for instance, is that they can be also not symmetric. Let X i = (X i1 , . . . , X id ), i ∈ {1, . . . , n}, be a sample of independent and identically distributed random vectors with common distribution function F, univariate marginal distribution functions F1 , . . . , Fd , and copula C F . For convenience, assume F is continuous. Consider the vector of componentwise maxima: Mn = (Mn,1 , . . . , Mn,d ) where Mn, j =
n
Xi j .
(1)
i=1
with denoting maximum. Since the joint and marginal distribution functions of Mn are given by F n and F1n , . . . , Fdn , respectively, it follows that the copula Cn of Mn is given by 1/n
1/n
Cn (u 1 , . . . , u d ) = C F (u 1 , . . . , u d )n , (u 1 , . . . , u d ) ∈ [0, 1]d .
(2)
The family of extreme-value copulas arises as the limits of these copulas Cn as the sample size n tends to infinity. The following definitions and theorems are taken from [13]. Definition 2.1. A copula C: [0, 1]d → [0, 1] is called an extreme-value copula if there exists a copula C F such that 1/n
1/n
C F (u 1 , . . . , u d )n → C(u 1 , . . . , u d ), (n → ∞)
(3)
for all (u 1 , . . . , u d ) ∈ [0, 1]d . The copula C F is said to be in the domain of attraction of C. The representation of extreme-value copulas can be simplified using the concept of max-stability. Definition 2.2. A d-variate copula C is max-stable if it satisfies the relationship 1/m
C(u 1 , . . . , u d ) = C(u 1
1/m m
, . . . , ud
)
for every integer m≥1 and all (u 1 , . . . , u d ) ∈ [0, 1]d . From the previous definitions, it is easy to see that a max-stable copula is in its own domain of attraction and thus must be itself an extreme-value copula. The converse is true as well. Theorem 2.1. A copula is an extreme-value copula if and only if it is max-stable. Denote as d−1 = {(w1 , . . . , wd ) ∈ [0, 1]d | j w j = 1} the unit simplex in R d . Then the class Ed of all d-dimensional extreme-value copulas is characterized as follows, see [13]. Theorem 2.2. A d-variate copula C is an extreme-value copula if and only if there exists a finite Borel measure H on d−1 , called spectral measure, such that C(u 1 , . . . , u d ) = exp(−l(− log u 1 , . . . , − log u d )), (u 1 , . . . , u d ) ∈ (0, 1]d ,
(4)
where the tail dependence function l: [0, ∞)d → [0, ∞) is given by l(x1 , . . . , xd ) =
d
(w j x j ) d H (w1 , . . . , wd ), (x1 , . . . , xd ) ∈ [0, ∞)d .
d−1 j=1
(5)
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The spectral measure H is arbitrary except for the d moment constraints w j d H (w1 , . . . , wd ) = 1, j ∈ {1, . . . , d}. d−1
(6)
The d moment constraints on H in (6) stem from the requirement that the margins of C be standard uniform. They imply that H (d−1 ) = d. Moreover, the fact that the class of spectral measures constrained by (6) is convex ensures the convexity of the class of all tail dependence functions Td (considering a fixed dimension d≥2). By a linear expansion of the logarithm and the exponential function, the domain of attraction equation (3) is equivalent to lim t −1 (1 − C F (1 − t x1 , . . . , 1 − t xd )) = − log C(e−x1 , . . . , e−xd ) = l(x1 , . . . , xd )
t→0
(7)
for all (x 1 , . . . , xd ) ∈ [0, ∞)d ; see for instance [6]. The tail dependence function l in (4) is convex, homogenous of order one , that is l(cx1 , . . . , cxd ) = cl(x1 , . . . , xd ) for c > 0, and satisfies max{x1 , . . . , xd }≤l(x1 , . . . , xd )≤x1 +· · ·+xd for all (x1 , . . . , xd ) ∈ [0, ∞)d . By homogeneity, it is characterized by the Pickands dependence function A: d−1 → [1/d, 1], which is the restriction of l to the unit simplex: l(x1 , . . . , xd ) = (x1 + · · · + xd )A(w1 , . . . , wd ) where w j =
xj , x1 + · · · + xd
(8)
for (x1 , . . . , xd ) ∈ [0, ∞)d \{0}d hence A can be considered as a function of d − 1 variables. The extreme-value copula C can be expressed in terms of A via ⎧⎛ ⎞ ⎫ d ⎬ ⎨ log u 1 log u d . (9) log u j ⎠ A d , . . . , d C(u 1 , . . . , u d ) = exp ⎝ ⎩ log u j log u j ⎭ j=1
j=1
j=1
Observe that if d = 2, then the Pickands dependence function A: 1 → [ 21 , 1] can be seen as a function of one variable, A(w1 , 1 − w1 ) = A(w1 ) (keeping the same notation A). Then we have the next simplified characterization of the class E2 of two-dimensional extreme-value copulas. Theorem 2.3. A bivariate copula C is an extreme-value copula if and only if C(u, v) = (uv) A(log u/ log(uv)) , (u, v) ∈ (0, 1)2
(10)
where A: [0, 1] → [1/2, 1] is convex and satisfies t ∨ (1 − t)≤A(t)≤1 for all t ∈ [0, 1]. Note that any d-dimensional Pickands dependence function A: d−1 → [1/d, 1] is convex and satisfies max{w1 , . . . , wd }≤A(w1 , . . . , wd )≤1 for all (w1 , . . . , wd ) ∈ d−1 . While this characterization completely determines Pickands dependence functions if d = 2, for d > 2 this is no longer true as the sufficiency is not guaranteed. Remark 2.1. (i) Consider a d-dimensional extreme value distribution function F : R d → [0, 1] with the marginal distribution functions F1 , . . . , Fd : R → [0, 1]. Denote by F1−1 , . . . , Fd−1 the corresponding quantile functions. Based on (4) and Sklar’s theorem, the corresponding tail dependence function l is given by l(x 1 , . . . , xd ) = − log F(F1−1 (exp(−x1 ), . . . , Fd−1 (exp(−xd )))). For more details see also [4]. (ii) As already mentioned, each tail dependence function l : [0, ∞]d → [0, ∞] satisfies, see [4, p. 257]: (L1) l(cx1 , . . . , cxd ) = cl(x1 , . . . , xd ) for all c ∈ (0, ∞) (homogeneity); (L2) l(e vector in R d ; di ) = 1 for the i-th unit d xi ; (L3) i=1 xi ≤l(x1 , . . . , xd )≤ i=1 (L4) l is convex function. (obviously, (L2) is a consequence of (L3)).
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If d = 2 then (L1)–(L4) are also sufficient conditions for a function l to be a tail dependence function. However, this is not the case if d > 2. As an example, see [4, p. 257], consider l : [0, ∞)3 → [0, ∞ given by l(x 1 , x2 , x3 ) = (x1 + x2 ) ∨ (x2 + x3 ) ∨ (x3 + x1 ). Evidently, l satisfies (L1)–(L4). Applying the formula (4), we can construct a function C : [0, 1]3 → [0, 1] given by C(u 1 , u 2 , u 3 ) = min{u 1 u 2 , u 2 u 3 , u 3 u 1 }. However the C-volume of the cube [ 23 , 1]3 is − 19 , i.e., C is not a copula. Now we turn our attention to the class of Archimedean copulas, which are linked to the continuous Archimedean t-norms [1,19]. Indeed, a d-dimensional copula C: [0, 1]d → [0, 1] is called Archimedean, C ∈ Ad , whenever d −1 min f (0), C(u 1 , . . . , u n ) = f f (u i ) (11) i=1
for some function f : [0, 1] → [0, ∞]. Then f is called an additive generator of C and it is unique up to a positive multiplicative constant. Due to Moynihan [26] it is known that additive generator of bivariate Archimedean copulas are characterized by the convexity, strict monotonicity, continuity and f (1) = 0. For d > 2, the next characterization was shown by Mc Neil and Nešlehová [24]. Theorem 2.4. Let f : [0, 1] → [0, ∞] be an additive generator of a bivariate Archimedean copula and let d ∈ N , d > 2 be fixed. Then the following are equivalent. (i) the function C: [0, 1]d → [0, 1] given by (11) is an Archimedean copula, C ∈ Ad . (ii) the function g: [−∞, 0] → [0, 1], g(x) = f −1 (min{ f (0), −x}) has non-negative derivatives of order 1, . . . , d − 2 on (−∞, 0), and g (d−2) is convex. We denote the class of all additive generators of Archimedean copulas from Ad by Gd . Observe that the weakest 1), and its Archimedean copula Cd ∈ Ad is a member of non-strict Clayton family with parameter = −1/(d − d (1 − additive generator is given by f dw (x) = 1 − x 1/(d−1) . Consequently Cdw (u 1 , . . . , u d ) = (max{0, 1 − i=1 1/(d−1) d−1 )}) . Note also that due to Kimberling [18], an additive generator f : [0, 1] → [0, ∞] generates an ui Archimedean copula C via (11) for any arity d = 2, 3, . . . if and only if the related function g: [−∞, 0] → [0, 1] is absolutely monotone (i.e., it has all derivatives on (−∞, 0) and these derivatives are non-negative). The weakest additive generator of this type is given by f (x) = − log x and it generates the product copula , see [27]. 3. Dependence functions We recall first some well known families of dependence functions. Example 3.1. (i) A parametric class (l )∞ =1 ⊂ Td of logistic tail dependence functions (linked to logistic copulas, called also Gumbel–Hougaard copulas) is given by ⎧ 1/ ⎪ d ⎨ xi if ∈ [1, ∞), l (x1 , . . . , xd ) = (12) ⎪ ⎩ i=1 max{x1 , . . . , xd } if = ∞. and thus also the corresponding copulas C : [0, 1]d → [0, 1] given by C (u 1 , . . . , u d ) Observe that dl are symmetric 1/ = exp(−( i=1 (− log u i ) ) ) if ∈ [1, ∞), and C∞ (u 1 , . . . , u d ) = min{u 1 , . . . , u d } are symmetric. d u a(i)i , where u (i) is the i-th (ii) A well known class of Cuadras–Augé copulas is given by Ca (u 1 , . . . , u d ) = i=1 order statistics of the sample (u 1 , . . . , u d ) (i.e., u (1) = min{u 1 , . . . , u d }, while u (d) = max{u 1 , . . . , u d }), and the d is d-monotone, i.e., each parameters vector a = (a1 , . . . , ad ) satisfies a1 = 1, ad ≥0 and the sequence (ai )i=1 m For more details see [22]. The corresponding tail its difference of order m ∈ {1, . . . , d − 1} has sign (−1) . d dependence function la ∈ Td are given by la (x1 , . . . , xd ) = i=1 ai x(d−i+1) .
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Another important subclass of extreme-value copulas is formed by Marshall–Olkin copulas (for a detailed discussion see, e.g., [21]). As an example, for d = 2, the bivariate Marshall–Olkin copulas are determined by two parameters , ∈ [0, 1] and the corresponding tail dependence functions are given by l, (x1 , x2 ) = max{x1 + x2 , x1 +(1−)x2 }. In the next example, an interesting three-dimensional dependence function (in both forms) is recalled. Example 3.2. Fix d = 3 and spectral measure H : 2 → [0, ∞[, H = (1/2,1/2,0) +(1/2,0,1/2) +(0,1/2,1/2) , ((w1 ,w2 ,w3 ) is the Dirac measure concentrated in point (w1 , w2 , w3 ) ∈ 2 ). Then the corresponding tail dependence function 3 ¯ 1 , x2 , x3 ) = i=1 ¯ [0, ∞)3 → [0, ∞) is given by l(x xi + 21 med(x1 , x2 , x3 ). The related Pickands dependence l: 3 ¯ 2 → [0, 1] is given by A(w ¯ 1 , w2 , w3 ) = l(w ¯ 1 , w2 , w3 ) = i=1 function A: wi + 21 med(w1 , w2 , w3 ), and the √ corresponding extreme-value copula Cl¯ ∈ C3 is given by Cl¯(u 1 , u 2 , u 3 ) = min{u 1 , u 2 , u 3 } med(u 1 , u 2 , u 3 ). Evidently, Cl¯ is symmetric. In the next theorem we introduce a new class of dependence functions. Theorem 3.1. Fix d ∈ N , d≥2, and a partition P = {B1 , . . . , Bk } of the set {1, . . . , d}. Then the mapping HP : d−1 → [0, ∞[ given by HP =
k
|B j |[B j ] ,
(13)
j=1
where |B j | is the cardinality of the set B j , and [B j ] is the Dirac measure concentrated in the point [B j ] = (b j1 , . . . , b jd ) ∈ d−1 , ⎧ ⎨ 1 i f i ∈ Bj, b ji = |B j | ⎩ 0 else is a spectral measure. The corresponding tail dependence function lP ∈ Td is given by lP (x1 , . . . , xd ) = kj=1 max {xi |i ∈ B j }, and the related Pickands dependence function AP is given by AP (w1 , . . . , wd ) = kj=1 max{wi |i ∈ B j }. Moreover, the corresponding extreme-value copula CP ∈ E d is given by CP (u 1 , . . . , u d ) = kj=1 min{u i |i ∈ B j }. Proof. Due to (13), it follows that for i ∈ B j d−1 wi d H (w1 , . . . , wd ) = 1, i ∈ 1, . . . , d. As for each i ∈ {1, . . . , d} there is j ∈ {1, . . . , k} such that i ∈ B j , HP is a spectral measure. Following (5), we get that lP (x1 , . . . , xd ) = k d d i=1 (wi x i ) d HP (w1 , . . . , wd ) = d−1 j=1 (w j x j ) d H (w1 , . . . , wd ) = j=1 ( i∈B j ((1/|B j |)x i )|B j |) = d−1 k j=1 max{x i |i ∈ B j }. The rest of the proof is a matter of an easy calculation. Observe that our parametric class of extreme value copulas contains both boundary members (related to the finest partition P∗ = {{1}, . . . , {d}}) and Min (related to the coarsest partition P∗ = {1, . . . , d}). In general, the copula CP is related to a random vector formed by independent sub-vectors, each of them having a comonotone copula. Note that the construction of dependence functions for bivariate copulas (i.e., when d = 2) by means of Dirac measures was discussed also in [23]. Denote by Td the class of all d-dimensional tail dependence functions, and by Pd the class of all d-dimensional Pickands dependence functions. As already mentioned, due to the convexity of the class of all spectral measures of dimension d, see Theorem 2.2, both classes Td and Pd are convex, i.e., they are closed under convex sums. We introduce a method for constructing new members of Td (Pd ) generalizing the convexity-based approach. Note, first of all, that for any d-dimensional copulas C1 , . . . , Cn , due to [20] also the function C: [0, 1]d → [0, 1] given by C(u 1 , . . . , u d ) =
n
C j (u 1 j1 , . . . , u d jd ),
j=1
where ji ≥0 and
n
i=1 ji
= 1, j ∈ {1, . . . , n}, i ∈ {1, . . . , d}, is a member of Cd .
(14)
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Moreover, evidently if C1 , . . . , Cn ∈ Ed then also C ∈ Ed . This observation is a background of the next construction method for d-dimensional tail dependence functions. Theorem 3.2. Let l1 , . . . , ln ∈ Td and let ji ≥0 and l: [0, ∞)d → [0, ∞) given by l(x1 , . . . , xd ) =
n
n
i=1 ji
= 1, j ∈ {1, . . . , n}, i ∈ {1, . . . , d}. Then the function
l j ( j1 x1 , . . . , jd xd )
(15)
j=1
is also a tail dependence function, l ∈ Td . Moreover, if ji = j , j ∈ {1, . . . , n}, i ∈ {1, . . . , d}, then l = is the standard convex sum.
n
j=1 j l j
Proof. For fixed l1 , . . . , ln ∈ Td , let C j : [0, 1]d → [0, 1], j = {1, . . . , n} be given by (4). Due to Theorem 2.2., each C j ∈ Ed , and thus also C ∈ Ed , where C is given by (14). Let l: [0, ∞)d → [0, ∞) be the corresponding tail dependence function linked to C. Then, for each (u 1 , . . . , u d ) ∈ (0, 1]d , it holds C(u 1 , . . . , u d ) = exp(−l(− log u 1 , . . . , − log u d )) =
n
C j (u 1 j1 , . . . , u d jd )
j=1
=
n
exp(−li (− log u 1 j1 , . . . , − log u d jd ))
j=1
⎛ = exp ⎝−
n
⎞ li (− j1 log u 1 , . . . , − jd log u d )⎠ .
j=1
= · · · = jd = j , j = Denoting − log u 1 = x1 , . . . , − log u d = xd , the result (15) follows. Moreover if j1 1, . . . , n, due to the homogeneity of tail dependence functions we have l(x 1 , . . . , xn ) = nj=1 l j ( j x1 , . . . , j xd ) = n j=1 j l j (x 1 , . . . , x d ). The above result can be rewritten for the Pickands dependence functions as follows. n Corollary 3.1. Let A1 , . . . , An ∈ Pd and let ji ≥0 and i=1 ji = 1, j ∈ {1, . . . , n}, i ∈ {1, . . . , d}. Then the function A: d−1 → [1/d, 1] given by d n jd wd j1 w1 A(w1 , . . . , wd ) = ji wi A j d , . . . , d i=1 ji wi i=1 ji wi j=1 i=1 is a Pickands dependence function, A ∈ Pd . Moreover, if ji = j , j ∈ {1, . . . , n}, i ∈ {1, . . . , d}, then A = nj=1 j A j is the standard convex sum. Proof. The result follows from (15). Indeed, for fixed A1 , . . . , An ∈ Pd , define l1 , . . . , ln : [0, ∞)d → d (8) and from d d xi ). Then [0, ∞) by l j (x1 , . . . , xd ) = ( i=1 xi )A j (x1 / i=1 xi , . . . , xd / i=1 A(w1 , . . . , wn ) = l(w1 , . . . , wn ) =
n
l j ( j1 w1 , . . . , jd wd )
j=1
=
d n j=1
i=1
ji wi
Aj
j1 w1
d
i=1 ji wi
jd wd
, . . . , d
i=1 ji wi
.
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Observe that for d = 2, n = 2, Corollary 3.1 was proved in [17], see also [11] where this method is called “Asymmetrization", while for d = 2 and n > 2 in [2]. Based on the dependence functions from Example 3.1, it holds: Example 3.3. (i) Applying Theorem 4.1 to l1 , . . . , ln , the tail dependence function l is not symmetric, in general. Fix 1 = · · · = n = ∞. Then l(x1 , . . . , xd ) = nj=1 max{ j1 x1 , . . . , jd xd }, and the corresponding extreme-value copula C ∈ Ed is given by C(u 1 , . . . , u d ) = nj=1 min{u 1 j1 , . . . , u d jd }. ¯ see Example 3.2, and vectors (1, 0), (1, 0) and ( 1 , 1 ). Applying Theorem 4.1, we get a new (ii) Consider l1 = l2 = l, 2 2 ˜ 1 , x2 , x3 ) = l(x ¯ 1 , 0, x3 /2) + l(0, ¯ x2 , x3 /2) = x1 ∨ x3 /2 + x2 ∨ x3 /2 + tail dependence function l˜ ∈ T3 given by l(x 1 1 x3 /2 + x2 ∨ x3 /2). The corresponding extreme-value copula 2 (x 1 ∧ x 3 /2 + x 2 ∧ x 3 /2) = 2 (x 1 + x 2 + x 3 + x 1 ∨√ √ Cl˜ ∈ E3 is given by Cl˜(u 1 , u 2 , u 3 ) = (u 1 u 2 u 3 (u 1 ∧ u 3 )(u 2 ∧ u 3 ))1/2 , and it is clearly not symmetric. 4. d-Dimensional Archimax copulas Two-dimensional Archimax copulas were introduced by Capéraá et al. [5] as a common extension of both extremevalue copulas and Archimedean copulas. Indeed, they have shown that, for any Pickands dependence function A ∈ P2 and any additive generator f ∈ G2 , the function f (u) −1 C f,A (u, v) = f min f (0), f (u) + f (v))A (16) f (u) + f (v) is a bivariate copula, C f,A ∈ C2 . Evidently (16) can be rewritten for the corresponding tail dependence function l ∈ T2 into C f,A (u, v) = C f,l (u, v) = f −1 (min{ f (0), l( f (u), f (v))}).
(17)
One of the aims of our paper is the proposal of d-dimensional Archimax copulas for d > 2. C f,l (u 1 , . . . , u d ) = f −1 (min{ f (0), l( f (u 1 ), . . . , f (u d ))}).
(18)
Definition 4.1. For a fixed d≥2, d ∈ N , l ∈ Td and f ∈ G2 , let C f,l given by (18) be a copula, C f,l ∈ Cd . Then C f,l is called a d-dimensional Archimax copula, and the class of all such copulas will be devoted by AMd . Due to [5], AM2 = {C f,l | f ∈ G2 , l ∈ T2 }. Clearly Ad ⊂AMd and Ed ⊂AMd . For some other d-dimensional Archimax copulas we can exploit the next result of Morillas [25]. Theorem 4.1. Fix d ∈ N , d≥2. Let C ∈ Cd and : [0, 1] → [0, 1] be a continuous strictly increasing function such that (1) = 1 and : [0, 1] → [0, 1] given by (x) = −1 (max{ (0), x}) has non-negative derivatives of orders 1, . . . , d on (0, 1). Then the function C : [0, 1]d → [0, 1] given by C (u 1 , . . . , u d ) = (C( (u 1 ), . . . , (u d )))
(19)
is a d-dimensional copula, C ∈ Cd . Due to Theorem 4.1, we can introduce the next subclass of d-dimensional Archimax copulas. Corollary 4.1. Fix d ∈ N , d > 2. Let l ∈ Td and : [0, 1] → [0, 1] satisfies the constraints of Theorem 5.1, then the function Cl, : [0, 1]d → [0, 1] given by Cl, (u 1 , . . . , u d ) = (exp(−l(− log (u 1 ), . . . , − log (u d ))))
(20)
is a d-dimensional Archimax copula, Cl, ∈ AMd . Proof. Recall that the product generated by − log x is a copula for any dimension d≥2 and that − log is an additive generator of a two-dimensional Archimedean copula because of [9]. Therefore, due to Theorem 4.1, − log generates a d-dimensional Archimedean copula, too. The result follows from Theorem 4.1.
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Remark 4.1. (i) The weakest d-dimensional Archimedean copula Cdw is also a member of AMd . However, it cannot be obtained by means of Corollary 5.1. (ii) Logistic tail dependence function l , see (12), can be combined with any additive generator f ∈ Gd . Indeed, then C f,l = C f is a d-dimensional Archimedean copula whenever ∈ [1, ∞[, and C f,l∞ = Min is the strongest d-dimensional copula. (iii) Tail dependence function l∞ ∈ Td can be combined with any additive generator f ∈ G2 (even f need not be convex!) and still C f,l∞ = Min ∈ Cd . As observed in Remark 5.1(iii), for some tail dependence functions l ∈ Td we can apply an additive generator f ∈ Gk , k < d, and still C f,l ∈ Cd . In what follows we will denote G1 = { f : [0, 1] → [0, ∞]| f is continuous, strictly decreasing and f (1) = 0}. Based on Theorem 3.1, we can introduce the next class of Archimax copulas. of {1, . . . , d} and let f ∈ C1 . Then the function Theorem 4.2. For fixed d≥2, let P = {B1 , . . . , Bk } be a partition C f,P : [0, 1]d → [0, 1] given by C f,P (u 1 , . . . , u d ) = f (−1) ( kj=1 f (min{u i |i ∈ B j })) is a d-dimensional copula if and only if f ∈ Gk . Proof. To see the necessity, observe that fixing (d − k) coordinates to be equal to 1, each k-dimensional marginal function of a d-dimensional copula is k-dimensional copula. For each j = 1, . . . , k, choose an index i j ∈ B j and for i ∈ / {i 1 , . . . , i k } fix u i = 1. Then the function C : [0, 1]k → [0, 1] given by C(u i1 , . . . , u ik ) = C f,P (u 1 , . . . , u d ) = (−1) ( kj=1 f (u i j )) is a k-dimensional Archimedean copula generated by f. Due to McNeil and Nešlehová results from f [24], f ∈ Gk . To show the sufficiency, one should prove that for each input vectors u, v ∈ [0, 1]d , u≤v, VC f,P ([u, v]) = I I I (−1) C f,P (z )≥0, where I ⊆ {1, . . . , d} and u i if i ∈ I, I zi = vi else. Obviously, if k = d, i.e., |B j | = 1 for each j = 1, . . . , d, then C f,P = C f is a d-dimensional Archimedean copula and thus f ∈ Gk . Suppose k < d. Then there is some j0 such that |B j0 | > 1. Choose m ∈ B j0 such that u m = min(u i |i ∈ B j0 ). Then for any I, J ⊆ {1, . . . , d} such that m ∈ I ∩ J and I \B j0 = J \B j0 , the equality C f,P (z I ) = C f,P (z J ) holds. Consequently VC f,P ([u,v]) = VC f,P ([u ,v ]) , where P = {B1 , . . . , B j0 −1 , B j0 \{m}, B j0 +1 , . . . , Bk } is a partition of , vm+1 , . . . , vd ), denoting vi = vi if i ∈ / B j0 , {1, . . . , d}\{m}, u = (u 1 , . . . , u m−1 , u m+1 , . . . , u d ), v = (v1 , . . . , vm−1 and vi = vi ∧ vm if i ∈ B j0 . If P consists of singletons only, i.e., k = d − 1, then f ∈ Gk ensures VC f,P [u,v] = VC f,P ([u ,v ]) ≥0. If not, we can continue by induction repeating the above arguments, coming to the fact the f ∈ Gk ensures VC f,P ([u,v]) ≥0, i.e., to the fact the C f,P is a d-dimensional copula. Note that the sufficiency in Theorem 4.2 can be shown using probabilistic arguments, too. Indeed, take U1 , . . . , Ud random variables that are uniformly distributed on [0, 1]. Take V1 , . . . , Vk random variables with joint distribution function given by the Archimedean copula C f . Suppose that, if there exists j such that s, t ∈ B j , then Us = Ut = V j almost surely, otherwise Us and Ut are independent. Then the joint distribution function of U1 , . . . , Ud is given by the copula considered in Theorem 4.2. Example 4.1. Consider a two member partition P B = {B, B c }, B ∅, B {1, . . . , d}. Then for any f ∈ G2 , C f,P : [0, 1]d → [0, 1] given by C f,P (u 1 , . . . , u d ) = f (−1) ( f (min{u i |i ∈ B}) + f (min{u i |i ∈ B}) is a d-dimensional copula. Consider the additive generator f W of the weakest 2-copula W, f W (x) = 1 − x. Then C f W ,P B (u 1 , . . . , u d ) = / B} − 1, 0} is a singular d-dimensional copula with support uniformly distributed on max{min{u i |i ∈ B} + min{u i |i ∈
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the segment connecting points 1 B and 1 B c , recall that 1 if i ∈ B, 1 B (i) = 0 else. Remark 4.2. (i) From probabilistic point of view, copula C f,P discussed in Theorem 4.2 is related to a random vector (U1 , . . . , Ud ) of random variables U1 , . . . , Ud uniformly distributed over [0, 1] such that each sub-vector (Ui |i ∈ B j ), j = 1, . . . , k is comonotone and all the sub-vectors are coupled by a k-dimensional Archimedean copula. (ii) The class of d-dimensional tail dependence functions is convex. However though two tail dependence functions l1 , l2 can be combined with an additive generator f ∈ Gk , k < d, so that C f,l1 and C f,l2 are d-dimensional copulas, this fact does not mean that also the convex combination l = l1 + (1 − )l2 can be combined with f so that C f,l is a d-dimensional copula. As an example it is enough to consider l1 (x1 , x2 , x3 ) = x1 + max{x2 , x3 } and l2 (x1 , x2 , x3 ) = x3 + max{x1 , x2 }. As already shown, C f w,l1 (u 1 , u 2 , u 3 ) = max{u 1 + min{u 2 , u 3 } − 1, 0} and C f w,l2 (u 1 , u 2 , u 3 ) = max{u 3 + min{u 1 , u 2 } − 1, 0} are copulas from C3 . Consider l = (l1 + l2 )/2, i.e., l(x1 , x2 , x3 ) = 21 (x1 + x3 + max{x2 , x3 } + max{x1 , x2 }). Then the function C f w,l : [0, 1]3 → [0, 1] given by C fw,l (u 1 , u 2 , u 3 ) = max{ 21 (u 1 + u 3 + min{u 2 + u 3 } + min{u 1 + u 2 }) − 1, 0} is not a copula. Indeed, the volume VC fw,l ([ 21 , 41 , 21 ] × [1, 1, 1]) = 1 − ( 21 + 41 + 21 ) = − 41 is negative. (iii) Note that some related ideas concerning Example 4.1 in the language of distribution functions can be found also in [28]. Inspired by the characterization of two-dimensional Archimax copulas, we finally open the next problem. Open Problem 4.1. Is it true that for any fixed d≥2, f ∈ Gd and l ∈ Td the function C f,l is a d-dimensional copula, i.e., C f,l ∈ AMd ? 5. Concluding remarks After discussion and introduction of new types of dependence functions and construction of dependence functions, we have introduced Archimax copulas of higher dimensions. These copulas are asymmetric, in general (up to special cases related to symmetric dependence functions) and they allow an interesting generalization of Marshall–Olkin copulas (which are extreme-value copulas and thus related to the additive generator f (x) = − log x). Moreover, they overlap with several other classes of multivariate copulas, such as vine copulas [3] (this happen for vine copulas linked to bivariate Archimax copulas based on a fixed additive generator) or hierarchial Archimedean copulas [16] (again a fixed additive generator of an Archimedea copula, or its powers, should be considered). Note that in both mentioned classes we are faced to similar problems ensuring the d-increasingness of introduced functions (hence to guarantee that they are d-dimensional copulas), and thus up to several sufficient conditions known for construction of these copulas there are several open problems. Concerning Archimax copulas we have introduced in this paper, one of such problems (solved for d = 2 in [5]) concerns the possible combination of any d-dimensional additive generator f ∈ Gd and any d-dimensional tail dependence function l ∈ Td , asking whether then C f,l is a d-dimensional copula. Acknowledgments The work on this contribution was supported by Grants APVV-LPP-0004-07, APVV-0496-10 and VEGA 1/0143/11. The authors are grateful to both anonymous referees and the area editor for all their comments and suggestions which helped us to improve significantly the original version of this paper, including, among others, an alternative probabilistic proof of Theorem 4.2. References [1] C. Alsina, M.J. Frank, B. Schweizer, Associative Functions: Triangular Norms and Copulas, World Scientific, Singapore, 2006. [2] T. Bacigál, V. Jágr, R. Mesiar, Non-exchangeable random variables, Archimax copulas and their fitting to real data, Kybernetika 47 (2011) 519–531.
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Fuzzy Sets and Systems 228 (2013) 78 – 87 www.elsevier.com/locate/fss
d-Dimensional dependence functions and Archimax copulas Radko Mesiar∗ Vladimír Jágr Slovak University of Technology in Bratislava, Slovakia Available online 2 November 2012
Abstract We introduce a new class of dependence functions and method for constructing tail and Pickands dependence functions. Finally, we conjecture the structure of d-dimensional Archimax copulas and discuss some classes of d-dimensional Archimax copulas. © 2012 Elsevier B.V. All rights reserved. Keywords: Copula; Extreme value copula; Archimedean copula; Archimax copula; Tail dependence function; Pickands dependence function
1. Introduction We suppose the reader to be familiar with the basic definitions and properties of copulas. For an overview for unfamiliar readers we recommend the monographs [12,15,16,27], a recent special issue of Information Sciences [8], and the recent papers [10,14]. While the case of two-dimensional copulas is discussed in many papers and books, including the proposal of several classes of two-dimensional copulas and construction methods for them, there are only few known classes of d-dimensional copulas with d > 2. We denote by Cd the set of all d-dimensional copulas. Recall that Cd is a convex and compact set (in the uniform convergence topology) of functions. Two distinguished classes in Cd are the extreme value copulas Ed (related to tail dependence functions, which when normed are called Pickands dependence functions), and the Archimedean copulas Ad (related to additive generators). For d = 2, there is an interesting extension of both C2 and A2 , namely the class AM2 of two-dimensional Archimax copulas. The aim of this paper is to discuss d-dimensional dependence functions and d-dimensional Archimax copulas, i.e., the class AMd , d > 2. Note that the preliminary version of this paper was presented at the summer school AGOP’2011 in Benevento, and here we have added several new results. The paper is organized as follows. In the next section, we recall extreme value copulas and d-dimensional Archimedean copulas. Note that this section is taken mostly from [13,24]. In Section 3, some well known d-dimensional dependence functions are recalled and a new class of such functions is introduced. It includes also a proposal of a construction method for tail dependence functions and Pickands dependence functions. In Section 4 we recall first the class AM2 of two-dimensional Archimax copulas [5]. Then we define the class AMd of d-dimensional Archimax copulas d > 2, and introduce some subclasses of AMd . Moreover, we introduce d-dimensional copulas based on a new class of d-dimensional dependence functions given in Section 3 and on additive generators of k-dimensional Archimedean copulas with k < d. Finally, some concluding remarks are given. ∗ Corresponding author.
E-mail addresses: [email protected] (R. Mesiar), [email protected] (V. Jágr). 0165-0114/$ - see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.fss.2012.10.010
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2. Extreme value copulas and Archimedean copulas This section is based on an overview paper on extreme-value copulas [13] and on [24] devoted to the study of d-dimensional Archimedean copulas. Extreme-value copulas not only arise naturally in the domain of extreme events, but they can also be a convenient choice to model data with positive dependence. An advantage with respect to the much more popular class of Archimedean copulas, for instance, is that they can be also not symmetric. Let X i = (X i1 , . . . , X id ), i ∈ {1, . . . , n}, be a sample of independent and identically distributed random vectors with common distribution function F, univariate marginal distribution functions F1 , . . . , Fd , and copula C F . For convenience, assume F is continuous. Consider the vector of componentwise maxima: Mn = (Mn,1 , . . . , Mn,d ) where Mn, j =
n
Xi j .
(1)
i=1
with denoting maximum. Since the joint and marginal distribution functions of Mn are given by F n and F1n , . . . , Fdn , respectively, it follows that the copula Cn of Mn is given by 1/n
1/n
Cn (u 1 , . . . , u d ) = C F (u 1 , . . . , u d )n , (u 1 , . . . , u d ) ∈ [0, 1]d .
(2)
The family of extreme-value copulas arises as the limits of these copulas Cn as the sample size n tends to infinity. The following definitions and theorems are taken from [13]. Definition 2.1. A copula C: [0, 1]d → [0, 1] is called an extreme-value copula if there exists a copula C F such that 1/n
1/n
C F (u 1 , . . . , u d )n → C(u 1 , . . . , u d ), (n → ∞)
(3)
for all (u 1 , . . . , u d ) ∈ [0, 1]d . The copula C F is said to be in the domain of attraction of C. The representation of extreme-value copulas can be simplified using the concept of max-stability. Definition 2.2. A d-variate copula C is max-stable if it satisfies the relationship 1/m
C(u 1 , . . . , u d ) = C(u 1
1/m m
, . . . , ud
)
for every integer m≥1 and all (u 1 , . . . , u d ) ∈ [0, 1]d . From the previous definitions, it is easy to see that a max-stable copula is in its own domain of attraction and thus must be itself an extreme-value copula. The converse is true as well. Theorem 2.1. A copula is an extreme-value copula if and only if it is max-stable. Denote as d−1 = {(w1 , . . . , wd ) ∈ [0, 1]d | j w j = 1} the unit simplex in R d . Then the class Ed of all d-dimensional extreme-value copulas is characterized as follows, see [13]. Theorem 2.2. A d-variate copula C is an extreme-value copula if and only if there exists a finite Borel measure H on d−1 , called spectral measure, such that C(u 1 , . . . , u d ) = exp(−l(− log u 1 , . . . , − log u d )), (u 1 , . . . , u d ) ∈ (0, 1]d ,
(4)
where the tail dependence function l: [0, ∞)d → [0, ∞) is given by l(x1 , . . . , xd ) =
d
(w j x j ) d H (w1 , . . . , wd ), (x1 , . . . , xd ) ∈ [0, ∞)d .
d−1 j=1
(5)
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The spectral measure H is arbitrary except for the d moment constraints w j d H (w1 , . . . , wd ) = 1, j ∈ {1, . . . , d}. d−1
(6)
The d moment constraints on H in (6) stem from the requirement that the margins of C be standard uniform. They imply that H (d−1 ) = d. Moreover, the fact that the class of spectral measures constrained by (6) is convex ensures the convexity of the class of all tail dependence functions Td (considering a fixed dimension d≥2). By a linear expansion of the logarithm and the exponential function, the domain of attraction equation (3) is equivalent to lim t −1 (1 − C F (1 − t x1 , . . . , 1 − t xd )) = − log C(e−x1 , . . . , e−xd ) = l(x1 , . . . , xd )
t→0
(7)
for all (x 1 , . . . , xd ) ∈ [0, ∞)d ; see for instance [6]. The tail dependence function l in (4) is convex, homogenous of order one , that is l(cx1 , . . . , cxd ) = cl(x1 , . . . , xd ) for c > 0, and satisfies max{x1 , . . . , xd }≤l(x1 , . . . , xd )≤x1 +· · ·+xd for all (x1 , . . . , xd ) ∈ [0, ∞)d . By homogeneity, it is characterized by the Pickands dependence function A: d−1 → [1/d, 1], which is the restriction of l to the unit simplex: l(x1 , . . . , xd ) = (x1 + · · · + xd )A(w1 , . . . , wd ) where w j =
xj , x1 + · · · + xd
(8)
for (x1 , . . . , xd ) ∈ [0, ∞)d \{0}d hence A can be considered as a function of d − 1 variables. The extreme-value copula C can be expressed in terms of A via ⎧⎛ ⎞ ⎫ d ⎬ ⎨ log u 1 log u d . (9) log u j ⎠ A d , . . . , d C(u 1 , . . . , u d ) = exp ⎝ ⎩ log u j log u j ⎭ j=1
j=1
j=1
Observe that if d = 2, then the Pickands dependence function A: 1 → [ 21 , 1] can be seen as a function of one variable, A(w1 , 1 − w1 ) = A(w1 ) (keeping the same notation A). Then we have the next simplified characterization of the class E2 of two-dimensional extreme-value copulas. Theorem 2.3. A bivariate copula C is an extreme-value copula if and only if C(u, v) = (uv) A(log u/ log(uv)) , (u, v) ∈ (0, 1)2
(10)
where A: [0, 1] → [1/2, 1] is convex and satisfies t ∨ (1 − t)≤A(t)≤1 for all t ∈ [0, 1]. Note that any d-dimensional Pickands dependence function A: d−1 → [1/d, 1] is convex and satisfies max{w1 , . . . , wd }≤A(w1 , . . . , wd )≤1 for all (w1 , . . . , wd ) ∈ d−1 . While this characterization completely determines Pickands dependence functions if d = 2, for d > 2 this is no longer true as the sufficiency is not guaranteed. Remark 2.1. (i) Consider a d-dimensional extreme value distribution function F : R d → [0, 1] with the marginal distribution functions F1 , . . . , Fd : R → [0, 1]. Denote by F1−1 , . . . , Fd−1 the corresponding quantile functions. Based on (4) and Sklar’s theorem, the corresponding tail dependence function l is given by l(x 1 , . . . , xd ) = − log F(F1−1 (exp(−x1 ), . . . , Fd−1 (exp(−xd )))). For more details see also [4]. (ii) As already mentioned, each tail dependence function l : [0, ∞]d → [0, ∞] satisfies, see [4, p. 257]: (L1) l(cx1 , . . . , cxd ) = cl(x1 , . . . , xd ) for all c ∈ (0, ∞) (homogeneity); (L2) l(e vector in R d ; di ) = 1 for the i-th unit d xi ; (L3) i=1 xi ≤l(x1 , . . . , xd )≤ i=1 (L4) l is convex function. (obviously, (L2) is a consequence of (L3)).
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If d = 2 then (L1)–(L4) are also sufficient conditions for a function l to be a tail dependence function. However, this is not the case if d > 2. As an example, see [4, p. 257], consider l : [0, ∞)3 → [0, ∞ given by l(x 1 , x2 , x3 ) = (x1 + x2 ) ∨ (x2 + x3 ) ∨ (x3 + x1 ). Evidently, l satisfies (L1)–(L4). Applying the formula (4), we can construct a function C : [0, 1]3 → [0, 1] given by C(u 1 , u 2 , u 3 ) = min{u 1 u 2 , u 2 u 3 , u 3 u 1 }. However the C-volume of the cube [ 23 , 1]3 is − 19 , i.e., C is not a copula. Now we turn our attention to the class of Archimedean copulas, which are linked to the continuous Archimedean t-norms [1,19]. Indeed, a d-dimensional copula C: [0, 1]d → [0, 1] is called Archimedean, C ∈ Ad , whenever d −1 min f (0), C(u 1 , . . . , u n ) = f f (u i ) (11) i=1
for some function f : [0, 1] → [0, ∞]. Then f is called an additive generator of C and it is unique up to a positive multiplicative constant. Due to Moynihan [26] it is known that additive generator of bivariate Archimedean copulas are characterized by the convexity, strict monotonicity, continuity and f (1) = 0. For d > 2, the next characterization was shown by Mc Neil and Nešlehová [24]. Theorem 2.4. Let f : [0, 1] → [0, ∞] be an additive generator of a bivariate Archimedean copula and let d ∈ N , d > 2 be fixed. Then the following are equivalent. (i) the function C: [0, 1]d → [0, 1] given by (11) is an Archimedean copula, C ∈ Ad . (ii) the function g: [−∞, 0] → [0, 1], g(x) = f −1 (min{ f (0), −x}) has non-negative derivatives of order 1, . . . , d − 2 on (−∞, 0), and g (d−2) is convex. We denote the class of all additive generators of Archimedean copulas from Ad by Gd . Observe that the weakest 1), and its Archimedean copula Cd ∈ Ad is a member of non-strict Clayton family with parameter = −1/(d − d (1 − additive generator is given by f dw (x) = 1 − x 1/(d−1) . Consequently Cdw (u 1 , . . . , u d ) = (max{0, 1 − i=1 1/(d−1) d−1 )}) . Note also that due to Kimberling [18], an additive generator f : [0, 1] → [0, ∞] generates an ui Archimedean copula C via (11) for any arity d = 2, 3, . . . if and only if the related function g: [−∞, 0] → [0, 1] is absolutely monotone (i.e., it has all derivatives on (−∞, 0) and these derivatives are non-negative). The weakest additive generator of this type is given by f (x) = − log x and it generates the product copula , see [27]. 3. Dependence functions We recall first some well known families of dependence functions. Example 3.1. (i) A parametric class (l )∞ =1 ⊂ Td of logistic tail dependence functions (linked to logistic copulas, called also Gumbel–Hougaard copulas) is given by ⎧ 1/ ⎪ d ⎨ xi if ∈ [1, ∞), l (x1 , . . . , xd ) = (12) ⎪ ⎩ i=1 max{x1 , . . . , xd } if = ∞. and thus also the corresponding copulas C : [0, 1]d → [0, 1] given by C (u 1 , . . . , u d ) Observe that dl are symmetric 1/ = exp(−( i=1 (− log u i ) ) ) if ∈ [1, ∞), and C∞ (u 1 , . . . , u d ) = min{u 1 , . . . , u d } are symmetric. d u a(i)i , where u (i) is the i-th (ii) A well known class of Cuadras–Augé copulas is given by Ca (u 1 , . . . , u d ) = i=1 order statistics of the sample (u 1 , . . . , u d ) (i.e., u (1) = min{u 1 , . . . , u d }, while u (d) = max{u 1 , . . . , u d }), and the d is d-monotone, i.e., each parameters vector a = (a1 , . . . , ad ) satisfies a1 = 1, ad ≥0 and the sequence (ai )i=1 m For more details see [22]. The corresponding tail its difference of order m ∈ {1, . . . , d − 1} has sign (−1) . d dependence function la ∈ Td are given by la (x1 , . . . , xd ) = i=1 ai x(d−i+1) .
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Another important subclass of extreme-value copulas is formed by Marshall–Olkin copulas (for a detailed discussion see, e.g., [21]). As an example, for d = 2, the bivariate Marshall–Olkin copulas are determined by two parameters , ∈ [0, 1] and the corresponding tail dependence functions are given by l, (x1 , x2 ) = max{x1 + x2 , x1 +(1−)x2 }. In the next example, an interesting three-dimensional dependence function (in both forms) is recalled. Example 3.2. Fix d = 3 and spectral measure H : 2 → [0, ∞[, H = (1/2,1/2,0) +(1/2,0,1/2) +(0,1/2,1/2) , ((w1 ,w2 ,w3 ) is the Dirac measure concentrated in point (w1 , w2 , w3 ) ∈ 2 ). Then the corresponding tail dependence function 3 ¯ 1 , x2 , x3 ) = i=1 ¯ [0, ∞)3 → [0, ∞) is given by l(x xi + 21 med(x1 , x2 , x3 ). The related Pickands dependence l: 3 ¯ 2 → [0, 1] is given by A(w ¯ 1 , w2 , w3 ) = l(w ¯ 1 , w2 , w3 ) = i=1 function A: wi + 21 med(w1 , w2 , w3 ), and the √ corresponding extreme-value copula Cl¯ ∈ C3 is given by Cl¯(u 1 , u 2 , u 3 ) = min{u 1 , u 2 , u 3 } med(u 1 , u 2 , u 3 ). Evidently, Cl¯ is symmetric. In the next theorem we introduce a new class of dependence functions. Theorem 3.1. Fix d ∈ N , d≥2, and a partition P = {B1 , . . . , Bk } of the set {1, . . . , d}. Then the mapping HP : d−1 → [0, ∞[ given by HP =
k
|B j |[B j ] ,
(13)
j=1
where |B j | is the cardinality of the set B j , and [B j ] is the Dirac measure concentrated in the point [B j ] = (b j1 , . . . , b jd ) ∈ d−1 , ⎧ ⎨ 1 i f i ∈ Bj, b ji = |B j | ⎩ 0 else is a spectral measure. The corresponding tail dependence function lP ∈ Td is given by lP (x1 , . . . , xd ) = kj=1 max {xi |i ∈ B j }, and the related Pickands dependence function AP is given by AP (w1 , . . . , wd ) = kj=1 max{wi |i ∈ B j }. Moreover, the corresponding extreme-value copula CP ∈ E d is given by CP (u 1 , . . . , u d ) = kj=1 min{u i |i ∈ B j }. Proof. Due to (13), it follows that for i ∈ B j d−1 wi d H (w1 , . . . , wd ) = 1, i ∈ 1, . . . , d. As for each i ∈ {1, . . . , d} there is j ∈ {1, . . . , k} such that i ∈ B j , HP is a spectral measure. Following (5), we get that lP (x1 , . . . , xd ) = k d d i=1 (wi x i ) d HP (w1 , . . . , wd ) = d−1 j=1 (w j x j ) d H (w1 , . . . , wd ) = j=1 ( i∈B j ((1/|B j |)x i )|B j |) = d−1 k j=1 max{x i |i ∈ B j }. The rest of the proof is a matter of an easy calculation. Observe that our parametric class of extreme value copulas contains both boundary members (related to the finest partition P∗ = {{1}, . . . , {d}}) and Min (related to the coarsest partition P∗ = {1, . . . , d}). In general, the copula CP is related to a random vector formed by independent sub-vectors, each of them having a comonotone copula. Note that the construction of dependence functions for bivariate copulas (i.e., when d = 2) by means of Dirac measures was discussed also in [23]. Denote by Td the class of all d-dimensional tail dependence functions, and by Pd the class of all d-dimensional Pickands dependence functions. As already mentioned, due to the convexity of the class of all spectral measures of dimension d, see Theorem 2.2, both classes Td and Pd are convex, i.e., they are closed under convex sums. We introduce a method for constructing new members of Td (Pd ) generalizing the convexity-based approach. Note, first of all, that for any d-dimensional copulas C1 , . . . , Cn , due to [20] also the function C: [0, 1]d → [0, 1] given by C(u 1 , . . . , u d ) =
n
C j (u 1 j1 , . . . , u d jd ),
j=1
where ji ≥0 and
n
i=1 ji
= 1, j ∈ {1, . . . , n}, i ∈ {1, . . . , d}, is a member of Cd .
(14)
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83
Moreover, evidently if C1 , . . . , Cn ∈ Ed then also C ∈ Ed . This observation is a background of the next construction method for d-dimensional tail dependence functions. Theorem 3.2. Let l1 , . . . , ln ∈ Td and let ji ≥0 and l: [0, ∞)d → [0, ∞) given by l(x1 , . . . , xd ) =
n
n
i=1 ji
= 1, j ∈ {1, . . . , n}, i ∈ {1, . . . , d}. Then the function
l j ( j1 x1 , . . . , jd xd )
(15)
j=1
is also a tail dependence function, l ∈ Td . Moreover, if ji = j , j ∈ {1, . . . , n}, i ∈ {1, . . . , d}, then l = is the standard convex sum.
n
j=1 j l j
Proof. For fixed l1 , . . . , ln ∈ Td , let C j : [0, 1]d → [0, 1], j = {1, . . . , n} be given by (4). Due to Theorem 2.2., each C j ∈ Ed , and thus also C ∈ Ed , where C is given by (14). Let l: [0, ∞)d → [0, ∞) be the corresponding tail dependence function linked to C. Then, for each (u 1 , . . . , u d ) ∈ (0, 1]d , it holds C(u 1 , . . . , u d ) = exp(−l(− log u 1 , . . . , − log u d )) =
n
C j (u 1 j1 , . . . , u d jd )
j=1
=
n
exp(−li (− log u 1 j1 , . . . , − log u d jd ))
j=1
⎛ = exp ⎝−
n
⎞ li (− j1 log u 1 , . . . , − jd log u d )⎠ .
j=1
= · · · = jd = j , j = Denoting − log u 1 = x1 , . . . , − log u d = xd , the result (15) follows. Moreover if j1 1, . . . , n, due to the homogeneity of tail dependence functions we have l(x 1 , . . . , xn ) = nj=1 l j ( j x1 , . . . , j xd ) = n j=1 j l j (x 1 , . . . , x d ). The above result can be rewritten for the Pickands dependence functions as follows. n Corollary 3.1. Let A1 , . . . , An ∈ Pd and let ji ≥0 and i=1 ji = 1, j ∈ {1, . . . , n}, i ∈ {1, . . . , d}. Then the function A: d−1 → [1/d, 1] given by d n jd wd j1 w1 A(w1 , . . . , wd ) = ji wi A j d , . . . , d i=1 ji wi i=1 ji wi j=1 i=1 is a Pickands dependence function, A ∈ Pd . Moreover, if ji = j , j ∈ {1, . . . , n}, i ∈ {1, . . . , d}, then A = nj=1 j A j is the standard convex sum. Proof. The result follows from (15). Indeed, for fixed A1 , . . . , An ∈ Pd , define l1 , . . . , ln : [0, ∞)d → d (8) and from d d xi ). Then [0, ∞) by l j (x1 , . . . , xd ) = ( i=1 xi )A j (x1 / i=1 xi , . . . , xd / i=1 A(w1 , . . . , wn ) = l(w1 , . . . , wn ) =
n
l j ( j1 w1 , . . . , jd wd )
j=1
=
d n j=1
i=1
ji wi
Aj
j1 w1
d
i=1 ji wi
jd wd
, . . . , d
i=1 ji wi
.
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Observe that for d = 2, n = 2, Corollary 3.1 was proved in [17], see also [11] where this method is called “Asymmetrization", while for d = 2 and n > 2 in [2]. Based on the dependence functions from Example 3.1, it holds: Example 3.3. (i) Applying Theorem 4.1 to l1 , . . . , ln , the tail dependence function l is not symmetric, in general. Fix 1 = · · · = n = ∞. Then l(x1 , . . . , xd ) = nj=1 max{ j1 x1 , . . . , jd xd }, and the corresponding extreme-value copula C ∈ Ed is given by C(u 1 , . . . , u d ) = nj=1 min{u 1 j1 , . . . , u d jd }. ¯ see Example 3.2, and vectors (1, 0), (1, 0) and ( 1 , 1 ). Applying Theorem 4.1, we get a new (ii) Consider l1 = l2 = l, 2 2 ˜ 1 , x2 , x3 ) = l(x ¯ 1 , 0, x3 /2) + l(0, ¯ x2 , x3 /2) = x1 ∨ x3 /2 + x2 ∨ x3 /2 + tail dependence function l˜ ∈ T3 given by l(x 1 1 x3 /2 + x2 ∨ x3 /2). The corresponding extreme-value copula 2 (x 1 ∧ x 3 /2 + x 2 ∧ x 3 /2) = 2 (x 1 + x 2 + x 3 + x 1 ∨√ √ Cl˜ ∈ E3 is given by Cl˜(u 1 , u 2 , u 3 ) = (u 1 u 2 u 3 (u 1 ∧ u 3 )(u 2 ∧ u 3 ))1/2 , and it is clearly not symmetric. 4. d-Dimensional Archimax copulas Two-dimensional Archimax copulas were introduced by Capéraá et al. [5] as a common extension of both extremevalue copulas and Archimedean copulas. Indeed, they have shown that, for any Pickands dependence function A ∈ P2 and any additive generator f ∈ G2 , the function f (u) −1 C f,A (u, v) = f min f (0), f (u) + f (v))A (16) f (u) + f (v) is a bivariate copula, C f,A ∈ C2 . Evidently (16) can be rewritten for the corresponding tail dependence function l ∈ T2 into C f,A (u, v) = C f,l (u, v) = f −1 (min{ f (0), l( f (u), f (v))}).
(17)
One of the aims of our paper is the proposal of d-dimensional Archimax copulas for d > 2. C f,l (u 1 , . . . , u d ) = f −1 (min{ f (0), l( f (u 1 ), . . . , f (u d ))}).
(18)
Definition 4.1. For a fixed d≥2, d ∈ N , l ∈ Td and f ∈ G2 , let C f,l given by (18) be a copula, C f,l ∈ Cd . Then C f,l is called a d-dimensional Archimax copula, and the class of all such copulas will be devoted by AMd . Due to [5], AM2 = {C f,l | f ∈ G2 , l ∈ T2 }. Clearly Ad ⊂AMd and Ed ⊂AMd . For some other d-dimensional Archimax copulas we can exploit the next result of Morillas [25]. Theorem 4.1. Fix d ∈ N , d≥2. Let C ∈ Cd and : [0, 1] → [0, 1] be a continuous strictly increasing function such that (1) = 1 and : [0, 1] → [0, 1] given by (x) = −1 (max{ (0), x}) has non-negative derivatives of orders 1, . . . , d on (0, 1). Then the function C : [0, 1]d → [0, 1] given by C (u 1 , . . . , u d ) = (C( (u 1 ), . . . , (u d )))
(19)
is a d-dimensional copula, C ∈ Cd . Due to Theorem 4.1, we can introduce the next subclass of d-dimensional Archimax copulas. Corollary 4.1. Fix d ∈ N , d > 2. Let l ∈ Td and : [0, 1] → [0, 1] satisfies the constraints of Theorem 5.1, then the function Cl, : [0, 1]d → [0, 1] given by Cl, (u 1 , . . . , u d ) = (exp(−l(− log (u 1 ), . . . , − log (u d ))))
(20)
is a d-dimensional Archimax copula, Cl, ∈ AMd . Proof. Recall that the product generated by − log x is a copula for any dimension d≥2 and that − log is an additive generator of a two-dimensional Archimedean copula because of [9]. Therefore, due to Theorem 4.1, − log generates a d-dimensional Archimedean copula, too. The result follows from Theorem 4.1.
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Remark 4.1. (i) The weakest d-dimensional Archimedean copula Cdw is also a member of AMd . However, it cannot be obtained by means of Corollary 5.1. (ii) Logistic tail dependence function l , see (12), can be combined with any additive generator f ∈ Gd . Indeed, then C f,l = C f is a d-dimensional Archimedean copula whenever ∈ [1, ∞[, and C f,l∞ = Min is the strongest d-dimensional copula. (iii) Tail dependence function l∞ ∈ Td can be combined with any additive generator f ∈ G2 (even f need not be convex!) and still C f,l∞ = Min ∈ Cd . As observed in Remark 5.1(iii), for some tail dependence functions l ∈ Td we can apply an additive generator f ∈ Gk , k < d, and still C f,l ∈ Cd . In what follows we will denote G1 = { f : [0, 1] → [0, ∞]| f is continuous, strictly decreasing and f (1) = 0}. Based on Theorem 3.1, we can introduce the next class of Archimax copulas. of {1, . . . , d} and let f ∈ C1 . Then the function Theorem 4.2. For fixed d≥2, let P = {B1 , . . . , Bk } be a partition C f,P : [0, 1]d → [0, 1] given by C f,P (u 1 , . . . , u d ) = f (−1) ( kj=1 f (min{u i |i ∈ B j })) is a d-dimensional copula if and only if f ∈ Gk . Proof. To see the necessity, observe that fixing (d − k) coordinates to be equal to 1, each k-dimensional marginal function of a d-dimensional copula is k-dimensional copula. For each j = 1, . . . , k, choose an index i j ∈ B j and for i ∈ / {i 1 , . . . , i k } fix u i = 1. Then the function C : [0, 1]k → [0, 1] given by C(u i1 , . . . , u ik ) = C f,P (u 1 , . . . , u d ) = (−1) ( kj=1 f (u i j )) is a k-dimensional Archimedean copula generated by f. Due to McNeil and Nešlehová results from f [24], f ∈ Gk . To show the sufficiency, one should prove that for each input vectors u, v ∈ [0, 1]d , u≤v, VC f,P ([u, v]) = I I I (−1) C f,P (z )≥0, where I ⊆ {1, . . . , d} and u i if i ∈ I, I zi = vi else. Obviously, if k = d, i.e., |B j | = 1 for each j = 1, . . . , d, then C f,P = C f is a d-dimensional Archimedean copula and thus f ∈ Gk . Suppose k < d. Then there is some j0 such that |B j0 | > 1. Choose m ∈ B j0 such that u m = min(u i |i ∈ B j0 ). Then for any I, J ⊆ {1, . . . , d} such that m ∈ I ∩ J and I \B j0 = J \B j0 , the equality C f,P (z I ) = C f,P (z J ) holds. Consequently VC f,P ([u,v]) = VC f,P ([u ,v ]) , where P = {B1 , . . . , B j0 −1 , B j0 \{m}, B j0 +1 , . . . , Bk } is a partition of , vm+1 , . . . , vd ), denoting vi = vi if i ∈ / B j0 , {1, . . . , d}\{m}, u = (u 1 , . . . , u m−1 , u m+1 , . . . , u d ), v = (v1 , . . . , vm−1 and vi = vi ∧ vm if i ∈ B j0 . If P consists of singletons only, i.e., k = d − 1, then f ∈ Gk ensures VC f,P [u,v] = VC f,P ([u ,v ]) ≥0. If not, we can continue by induction repeating the above arguments, coming to the fact the f ∈ Gk ensures VC f,P ([u,v]) ≥0, i.e., to the fact the C f,P is a d-dimensional copula. Note that the sufficiency in Theorem 4.2 can be shown using probabilistic arguments, too. Indeed, take U1 , . . . , Ud random variables that are uniformly distributed on [0, 1]. Take V1 , . . . , Vk random variables with joint distribution function given by the Archimedean copula C f . Suppose that, if there exists j such that s, t ∈ B j , then Us = Ut = V j almost surely, otherwise Us and Ut are independent. Then the joint distribution function of U1 , . . . , Ud is given by the copula considered in Theorem 4.2. Example 4.1. Consider a two member partition P B = {B, B c }, B ∅, B {1, . . . , d}. Then for any f ∈ G2 , C f,P : [0, 1]d → [0, 1] given by C f,P (u 1 , . . . , u d ) = f (−1) ( f (min{u i |i ∈ B}) + f (min{u i |i ∈ B}) is a d-dimensional copula. Consider the additive generator f W of the weakest 2-copula W, f W (x) = 1 − x. Then C f W ,P B (u 1 , . . . , u d ) = / B} − 1, 0} is a singular d-dimensional copula with support uniformly distributed on max{min{u i |i ∈ B} + min{u i |i ∈
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the segment connecting points 1 B and 1 B c , recall that 1 if i ∈ B, 1 B (i) = 0 else. Remark 4.2. (i) From probabilistic point of view, copula C f,P discussed in Theorem 4.2 is related to a random vector (U1 , . . . , Ud ) of random variables U1 , . . . , Ud uniformly distributed over [0, 1] such that each sub-vector (Ui |i ∈ B j ), j = 1, . . . , k is comonotone and all the sub-vectors are coupled by a k-dimensional Archimedean copula. (ii) The class of d-dimensional tail dependence functions is convex. However though two tail dependence functions l1 , l2 can be combined with an additive generator f ∈ Gk , k < d, so that C f,l1 and C f,l2 are d-dimensional copulas, this fact does not mean that also the convex combination l = l1 + (1 − )l2 can be combined with f so that C f,l is a d-dimensional copula. As an example it is enough to consider l1 (x1 , x2 , x3 ) = x1 + max{x2 , x3 } and l2 (x1 , x2 , x3 ) = x3 + max{x1 , x2 }. As already shown, C f w,l1 (u 1 , u 2 , u 3 ) = max{u 1 + min{u 2 , u 3 } − 1, 0} and C f w,l2 (u 1 , u 2 , u 3 ) = max{u 3 + min{u 1 , u 2 } − 1, 0} are copulas from C3 . Consider l = (l1 + l2 )/2, i.e., l(x1 , x2 , x3 ) = 21 (x1 + x3 + max{x2 , x3 } + max{x1 , x2 }). Then the function C f w,l : [0, 1]3 → [0, 1] given by C fw,l (u 1 , u 2 , u 3 ) = max{ 21 (u 1 + u 3 + min{u 2 + u 3 } + min{u 1 + u 2 }) − 1, 0} is not a copula. Indeed, the volume VC fw,l ([ 21 , 41 , 21 ] × [1, 1, 1]) = 1 − ( 21 + 41 + 21 ) = − 41 is negative. (iii) Note that some related ideas concerning Example 4.1 in the language of distribution functions can be found also in [28]. Inspired by the characterization of two-dimensional Archimax copulas, we finally open the next problem. Open Problem 4.1. Is it true that for any fixed d≥2, f ∈ Gd and l ∈ Td the function C f,l is a d-dimensional copula, i.e., C f,l ∈ AMd ? 5. Concluding remarks After discussion and introduction of new types of dependence functions and construction of dependence functions, we have introduced Archimax copulas of higher dimensions. These copulas are asymmetric, in general (up to special cases related to symmetric dependence functions) and they allow an interesting generalization of Marshall–Olkin copulas (which are extreme-value copulas and thus related to the additive generator f (x) = − log x). Moreover, they overlap with several other classes of multivariate copulas, such as vine copulas [3] (this happen for vine copulas linked to bivariate Archimax copulas based on a fixed additive generator) or hierarchial Archimedean copulas [16] (again a fixed additive generator of an Archimedea copula, or its powers, should be considered). Note that in both mentioned classes we are faced to similar problems ensuring the d-increasingness of introduced functions (hence to guarantee that they are d-dimensional copulas), and thus up to several sufficient conditions known for construction of these copulas there are several open problems. Concerning Archimax copulas we have introduced in this paper, one of such problems (solved for d = 2 in [5]) concerns the possible combination of any d-dimensional additive generator f ∈ Gd and any d-dimensional tail dependence function l ∈ Td , asking whether then C f,l is a d-dimensional copula. Acknowledgments The work on this contribution was supported by Grants APVV-LPP-0004-07, APVV-0496-10 and VEGA 1/0143/11. The authors are grateful to both anonymous referees and the area editor for all their comments and suggestions which helped us to improve significantly the original version of this paper, including, among others, an alternative probabilistic proof of Theorem 4.2. References [1] C. Alsina, M.J. Frank, B. Schweizer, Associative Functions: Triangular Norms and Copulas, World Scientific, Singapore, 2006. [2] T. Bacigál, V. Jágr, R. Mesiar, Non-exchangeable random variables, Archimax copulas and their fitting to real data, Kybernetika 47 (2011) 519–531.
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