ON A PROBLEM BY SCHWEIZER AND SKLAR

K Y B E R N E T I K A V O L U M E 4 8 ( 2 1 2 ), N U M B E R 2, P A G E S 2 8 7 2 9 3 ON A PROBLEM BY SCHWEIZER AND SKLAR Fabrizio Durante We give a representation of the class of all n dimensional copulas such that, for a fixed m N, 2 m < n, all their m dimensional margins are equal to the independence copula. Such an investigation originated from an open problem posed by Schweizer and Sklar. Keywords: Classification: copulas, distributions with given marginals, Fréchet Hoeffding bounds, partial mutual independence 6E5, 62E1 1. INTRODUCTION The representation and the construction of n dimensional distribution functions (=d.f. s) with given lower dimensional marginal distributions is one of the classical problem in probability theory, due to its relevance to applications. Questions of this kind arise, for example, when one wants to build a multivariate stochastic model and has some idea about the kind of dependence, or knows exactly certain marginal distributions (see, for instance, [1, 2, 3, 7, 9] and the references therein). In this note, we investigate a special problem of this type, namely we consider the class of all possible joint d.f. s of a random vector X = (X 1, X 2,, X n ) such that: (a) X i has a continuous d.f. F i, for each i {1, 2,, n}; (b) for a given m N, 2 m < n, every sub-vector of m elements in X is formed by independent random variables (=r.v. s). Such a problem has been originally posed by Schweizer and Sklar (see [1, Problem 6.7.3]) in the class of all distribution functions whose univariate margins are uniformly distributed on [, 1], i.e., in the class of copulas. In fact, in view of Sklar s Theorem [12], copulas are exactly the objects that allow to capture the dependence properties of a random vector. Therefore, in this note, we investigate the above-stated problem in terms of multivariate copulas and its lowerdimensional margins. The paper is organized as follows. First, in section 2 we define the basic elements that are necessary in order to make the paper self-contained. Then, in section 3, we characterize the dependence structures of the previous type by providing also some upper and lower bounds.

288 F. DURANTE 2. PRELIMINARIES Let n, m be in N, 2 m < n. We denote by P n,m the class of all permutations σ = (σ 1, σ 2,, σ n ) of (1, 2,, n) such that σ 1 < σ 2 < < σ m and σ m+1 < σ m+2 < < σ n. For example, P 3,2 = {(1, 2, 3), (1, 3, 2), (2, 3, 1)}. We denote by x = (x 1,, x n ) any point in R n and by I n the product of n copies of the unit interval I = [, 1]. For basic definitions and properties about copulas, we refer to [6, 8]. Here we recall that an n copula is a function C : I n I satisfying the following properties: (C1) C(u) = whenever u I n has at least one argument equal to ; (C2) C(u) = u i whenever u I n has all the arguments equal to 1 except possibly the ith one, which is equal to u i ; (C3) C is n increasing, viz., for each n box B = n [u i, v i ] I n, u i v i for any i {1, 2,, n}, V C (B) = sgn(z)c(z), z B where where the sum is taken over all vertices z in B, z i {u i, v i } for every i in {1, 2,, n}, and sgn(z) = 1, if the number of u i s among the arguments of z is odd, and sgn(z) = 1, otherwise. We denote by C n the set of all n copulas. For all C C n and for all u I n, W n (u) C(u) M n (u), (1) { n } W n (u) = max u i n + 1,, M n (u) = min{u 1, u 2,, u n }. These inequalities are called Fréchet Hoeffding bounds [7, 8]. Notice that M n C n, but W n C n only for n = 2. Another important n copula is Π n (u) = n u i, which is associated with independent r.v. s. Given C C n, the m marginals of C, 2 m < n, are the ( n m) m copulas obtained by setting (n m) of the arguments of C equal to 1. Moreover, we denote by C n (Π m ) the class of all n copulas such that all their m marginals are equal to Π m. Here, we present a method for constructing n copulas, which we shall use in the sequel. Proposition 2.1. Let C = {C t } t I m 1 t I m 1. Then C : I n I given by be a family in C n m+1 indexed by a parameter C(u) = um 1 is in C n, provided that the above integral exists. C t (u m,, u n ) dt 1 dt m 1. (2)

On a problem by Schweizer and Sklar 289 P r o o f. It is immediate to prove that the function C given by (2) satisfies (C1) and (C2). In order to prove that C is n increasing, consider the n box B = n [u i, v i ] in I n, u i v i for any i {1, 2,, n}. Then, we have that V C (B) = v1 u 1 vm 1 u m 1 V Ct ([u m, v m ] [u n, v n ]) dt 1 dt m 1, which is non negative because, for any t I m 1, C t belongs to C n m+1 and hence t V Ct ([u m, v m ] [u n, v n ]) is non negative. Example 2.2. Let C and {C t } t I m 1 be in C n m+1 and suppose that C t = C for every t I m 1. Then elementary integration yields ( m 1 ) D(u) = u i C(u m,, u n ), which is the n d.f. of the random vector (U 1, U 2,, U n ) such that: U i are uniformly distributed on I, C is the d.f. of (U m, U m+1,, U n ), Π m 1 is the d.f. of (U 1,, U m 1 ), and (U 1,, U m 1 ) and (U m,, U n ) are independent random vectors. Example 2.3. Let {C (t1,t 2)} (t1,t 2) I 2 be in C 2 defined by { Π 2 (u 1, u 2 ), t 1 1 C (t1,t 2)(u 1, u 2 ) = 2, M 2 (u 1, u 2 ), otherwise. Then, by using Proposition 2.1, we obtain that C 4 : I 4 I given by C 4 (u 1, u 2, u 3, u 4 ) = = u2 C (t1,t 2)(u 3, u 4 ) dt 1 dt 2 { Π 4 (u 1, u 2, u 3, u 4 ), u 1 1 2, u 2u 3u 4 2 + ( u 1 2) 1 u2 M 2 (u 3, u 4 ), otherwise, is a 4 copula such that all its 2 marginals are equal to Π 2. The copula C 4 can be also obtained by means of a gluing construction [11]. Remark 2.4. If C C n, then, for every permutation σ = (σ 1, σ 2,, σ n ) of (1, 2,, n) the function C σ : I n I given by C σ (u 1,, u n ) = C(u σ1,, u σn ) is also in C n. In particular, if C is the copula given by (2), then, for every permutation σ = (σ 1, σ 2,, σ n ) of (1, 2,, n), C σ : I n I given by is also in C n. C σ (u) = uσm 1 C t (u σm,, u σn ) dt 1 dt m 1 (3)

29 F. DURANTE 3. DESCRIPTION OF A SPECIAL CLASS OF COPULAS Following our approach, the description of the class of all possible joint d.f. s of a random vector X = (X 1, X 2,, X n ) such that, for a given m N, 2 m < n, every sub-vector of m elements in X is formed by independent r.v. s, is equivalent to the description of the class C n (Π m ). The elements of such a class are described in the following result. Theorem 3.1. Let n, m N, 2 m < n. The following statements are equivalent: (a) C C n (Π m ); (b) for every σ P n,m, there exists a family C σ = {Ct σ } t I m 1 for every u I n, in C n m+1 such that, C(u) = uσm 1 Ct σ (u σm,, u σn ) dt 1 dt m 1. (4) P r o o f. Let C be in C n (Π m ). Then, there exist a probability space (Ω, F, P) and a random vector U = (U 1, U 2,, U n ), U i uniformly distributed on I for every i {1, 2,, n}, such that C is the joint d.f. of U. Let σ P n,m. Then, for each u I n, C(u) = P(U 1 u 1,, U n u n ) = uσm 1 F σ t (u σm,, u σn ) dt 1 dt m 1, where, for every t = (t 1, t 2,, t m 1 ) I m 1, Ft σ : I n m+1 I defined by ( n ) Ft σ (u σm,, u σn ) = P {U σi u σi } U σ1 = t 1,, U σm 1 = t m 1, i=m is the (conditional) d.f. of (U σm,, U σn ) given (U σ1 = t 1,, U σm 1 = t m 1 ). The one dimensional marginals of Ft σ are uniformly distributed on I, because any subset of m elements in {U 1, U 2,, U n } is composed by independent r.v. s. Therefore, Ft σ is a copula and (b) follows. In the other direction, let C : I n I be such that, for every σ P n,m there exists a family C σ = {Ct σ } t I m 1 C n m+1 such that C can be represented in the form (4). Because of Proposition 2.1 (and Remark 2.4), C is a copula. Therefore, we have only to prove that all the m marginals of C are equal to Π m. To this end, let C m be the m marginal of C obtained by setting equal to 1 the arguments of C with indices ξ 1 < ξ 2 < < ξ n m, viz. C m (u 1,, u m ) = C(ũ), where ũ I n is obtained by setting ũ i = 1 for i {ξ 1,, ξ n m }, and ũ i = u i, otherwise. Consider the (unique) permutation ξ P n,m given by ξ = (ξ n m+1,, ξ n, ξ 1,, ξ n m ).

On a problem by Schweizer and Sklar 291 Then there exists a family C bξ ξ = {C b t } t I m 1 in C n m+1 such that uξn m+1 uξn 1 ξ C(u) = C b t (u ξ n, u ξ1, u ξn m ) dt 1 dt m 1. (5) ξ Since C b t satisfies (C2), equality (5) implies that Cm = Π m. For the arbitrariness of ξ 1, ξ 2,, ξ n m, it follows that C C n (Π m ). Remark 3.2. Since Theorem 3.1, if C C n (Π m ), then there exist ( n m) families C σ = {Ct σ } t I m 1 in C n m+1, each family associated with σ P n,m, such that C can be written in ( n m) different forms by means of (4). Moreover, for a fixed σ Pn,m, {Ct σ } t I m 1 is not uniquely determined: in fact, there exist infinitely many families D σ = {Dt σ } t I m 1 such that Ct σ Dt σ for every t belonging to a subset of I m 1 with (m 1) dimensional Lebesgue measure, and C can be represented in terms of D σ by means of (4). In the case n = 3 and m = 2, Theorem 3.1 can be reformulated in this form. Corollary 3.3. A 3 copula C 3 C 3 (Π 2 ) if, and only if, there exist three families of 2 copulas {C (1) t } t I, {C (2) t } t I and {C (3) t } t I, such that C 3 (u 1, u 2, u 3 ) = C (1) t (u 2, u 3 ) dt = u2 C (2) t (u 1, u 3 ) dt = u3 C (3) t (u 1, u 2 ) dt. In particular, we have that, for every i {1, 2, 3}, 1 C (i) t (u 1, u 2 ) dt = u 1 u 2. (6) A method for constructing families of 2 copulas that satisfy (6) is provided in [8, Example 3.1]. Specifically, for any 2 copula C, we can construct the family of 2 copula {C t } t I given by { C(1 t + u 1, u 2 ) C(1 t, u 2 ), u 1 t, C t (u 1, u 2 ) = u 2 C(1 t, u 2 ) + C(u 1 t, u 2 ), u 1 > t, which satisfies condition (6). Example 3.4. Let C θ be a member of the Eyraud Farlie Gumbel Morgenstern family of 3 copulas given by C θ (u 1, u 2, u 3 ) = u 1 u 2 u 3 (1 + θ(1 u 1 )(1 u 2 )(1 u 3 )), where θ [ 1, 1] (see [8]). Then C θ has all the 2 marginals equal to Π 2 and it can be expressed, for example, into the form C θ (u 1, u 2, u 3 ) = C σ t (u σ2, u σ3 ) dt,

292 F. DURANTE where σ = (σ 1, σ 2, σ 3 ) P 3,2, and C = {C σ t } t I is the family of 2 copulas given by for every t I and σ P 3,2. C σ t (u, v) = uv + θuv(1 u)(1 v)(1 2t), Theorem 3.1 can be rewritten in a simpler form if we suppose that C C n (Π m ) is exchangeable, viz. it does not change under permutation of its arguments. Corollary 3.5. Let n, m be in N, 2 m < n. Let C be an exchangeable copula. Then C C n (Π m ) if, and only if, there exists a family C = {C t } t I m 1 in C n m+1 such that, for every u I n, um 1 C(u) = C t (u m,, u n ) dt 1 dt m 1. (7) P r o o f. Let n, m be in N, 2 m < n. Let C be exchangeable. If C C n (Π m ), then Theorem 3.1 ensures that there exists a family C = {C t } t I m 1 in C n m+1 such that, C admits the representation (4). Conversely, if C can be represented in the form (7), then C(u 1,, u m, 1,, 1) = m u i, and, because C is exchangeable, all its m marginal d.f. s are equal to m u i, and, thus, C C n (Π m ). Pointwise upper and lower bounds for the class C n (Π m ) have been given in [4] (when n = 3 and m = 2, see also [5]). Theorem 3.1 provides also a way for obtaining them. In fact, for every σ P n,m there exists a family C σ = {Ct σ } t I m 1 in C n m+1 such that C C n can be represented in the form (4). Now, because every copula satisfies the inequalities (1), it follows that, for every u I n m+1 and for every t I m 1, W n m+1 (u) C σ t (u) M n m+1 (u). Thus, the following inequalities can be easily derived: where we define C L (u) = max σ P n,m C U (u) = min σ P n,m C L (u) C(u) C U (u), (8) {( m 1 {( m 1 u σi ) u σi ) W n m+1 (u σm,, u σn ) M n m+1 (u σm,, u σn ) An improvement of these bounds can be achieved by writing the expression for the survival d.f. associated with C and impose that it is non negative (see [7] for this procedure). } },.

On a problem by Schweizer and Sklar 293 ACKNOWLEDGEMENTS The support of School of Economics and Management, Free University of Bozen Bolzano, via the project Risk and Dependence is acknowledged. (Received August 4, 211) R E F E R E N C E S [1] V. Beneš and J. Štěpán, eds.: Distributions With Given Marginals and Moment Problems. Kluwer Academic Publishers, Dordrecht 1997. [2] C. M. Cuadras, J. Fortiana, and J. A. Rodriguez-Lallena, eds.: Distributions With Given Marginals and Statistical Modelling. Kluwer Academic Publishers, Dordrecht 22. Papers from the meeting held in Barcelona 2. [3] G. Dall Aglio, S. Kotz, and G. Salinetti, eds.: Advances in Probability Distributions with Given Marginals. Mathematics and its Applications 67, Kluwer Academic Publishers Group, Dordrecht 1991. Beyond the Copulas, Papers from the Symposium on Distributions with Given Marginals held in Rome 199. [4] P. Deheuvels: Indépendance multivariée partielle et inégalités de Fréchet. In: Studies in Probability and Related Topics, Nagard, Rome 1983, pp. 145 155. [5] F. Durante, E. P. Klement, and J. J. Quesada-Molina: Bounds for trivariate copulas with given bivariate marginals. J. Inequal. Appl. 28 (28), 1 9. [6] P. Jaworski, F. Durante, W. Härdle, and T. Rychlik, eds.: Copula Theory and its Applications. Lecture Notes in Statistics Proceedings 198, Springer, Berlin Heidelberg 21. [7] H. Joe: Multivariate Models and Dependence Concepts. Monographs on Statistics and Applied Probability 73, Chapman & Hall, London 1997. [8] R. B. Nelsen: An Introduction to Copulas. Second edition. Springer Series in Statistics, Springer, New York 26. [9] L. Rüschendorf, B. Schweizer, and M. D. Taylor, eds.: Distributions with Fixed Marginals and Related Topics. Institute of Mathematical Statistics, Lecture Notes Monograph Series 28, Hayward 1996. [1] B. Schweizer and A. Sklar: Probabilistic Metric Spaces. North-Holland Series in Probability and Applied Mathematics. North-Holland Publishing Co., New York 1983. Reprinted, Dover, Mineola 25. [11] K. F. Siburg and P. A. Stoimenov: Gluing copulas. Comm. Statist. Theory Methods 37 (28), 19, 3124 3134. [12] A. Sklar: Fonctions de répartition à n dimensions et leurs marges. Publ. Inst. Statist. Univ. Paris 8 (1959), 229 231. Fabrizio Durante, School of Economics and Management, Free University of Bozen Bolzano, Bolzano. Italy. e-mail: fabrizio.durante@unibz.it