Durante, Fabrizio

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Università del Salento: ESE - Salento University Publishing

Chapter 10Copulas and Schur–concavityThe notion of Schur–concavity (and the closely related concept of Schur–convexity)has a great importance in the recent applications of statistics; witness of this is therecent monograph of Spizzichino [152] where Schur–concavity is one of the centralthemes in the Bayesian models of aging. However, the study of Schur–concavity ofcopulas does not seem to have yet received any attention in the literature, althoughtwenty years ago Alsina studied the same question for t–norms (see [1]). To this topicthis chapter is devoted.In section 10.1 we present some results about the class of Schur–concave copulasand several examples are given in section 10.2. The concept of Schur–concavity,moreover, allows us to discuss an open problem on the classes of copulas and triangularnorms (section 10.3).The presented resulted are also contained in [44, 33].10.1 The class of Schur–concave copulasAt the beginning of the study on Schur–concavity of copulas, we recall some prop-erties that can be directly derived from section 1.2.Proposition 10.1.1. Let C : [0, 1]2 → [0, 1] be a semicopula.(a) If C is Schur–concave (or Schur-convex), then it is symmetric.(b) If C is Schur–concave (or Schur–convex) on ∆+ := {(x, y) ∈ [0, 1]2 : x ≥ y},then C is Schur–concave (or Schur–convex) on [0, 1]2.Proposition 10.1.2. A semicopula C : [0, 1]2 → [0, 1] is Schur–concave if, and onlyif, for all x, y and λ in [0, 1]C(x, y) ≤ C(λx+ (1− λ)y, (1− λ)x+ λy).136 Chap. 10 Copulas and Schur–concavityProof. It suffices to consider the definition of Schur–concavity and Corollary 1.2.1.Example 10.1.1. Consider the copulaM . For every x ≥ y, we have y ≤ λx+(1−λ)yand y ≤ (1− λ)x+ λy, so thatM(x, y) ≤M(λx+ (1− λ)y, (1− λ)x+ λy);and, analogously, we have the same result for x < y. Therefore M is Schur–concave.Proposition 10.1.3. Let C be a continuously differentiable semicopula. Then C isSchur–concave on [0, 1]2 if, and only if,(i) C is symmetric;(ii) for all (x, y) ∈ ∆+, ∂1C(x, y) ≤ ∂2C(x, y).As a consequence, it is easily proved that the copula Π is Schur–concave. Notethat not every symmetric copula is Schur–concave, as the following example shows.Example 10.1.2. Let C be the absolutely continuous copula defined byC(x, y) :=xy/2, if (x, y) ∈ [0, 1/2]× [0, 1/2];x (3y − 1)/2, if (x, y) ∈ [0, 1/2]× [1/2, 1];y (3x− 1)/2, if (x, y) ∈ [1/2, 1]× [0, 1/2];(xy + x+ y − 1) /2, if (x, y) ∈ [1/2, 1]× [1/2, 1].This copula is symmetric and has a density c given byc(x, y) :=1/2, if (x, y) ∈ [0, 1/2]2 ∪ [1/2, 1]2;3/2, otherwise.The three points x = (6/10, 4/10), y = (7/10, 3/10) and z = (8/10, 2/10) are suchthat x ≺ y ≺ z, butC(610,410)=32200<33200= C(710,310),C(710,310)=33200>28200= C(810,210).Therefore C is not Schur–concave.The following result allows us to investigate only on the class of Schur–concavecopulas.Proposition 10.1.4. The copula W is the only Schur–convex (quasi–)copula.Chap. 10 Copulas and Schur–concavity 137Proof. Let C be a Schur–convex copula. Given x, y ∈ [0, 1] such that x + y ≤ 1, wehave (x, y) ≺ (x+ y, 0), from whichC(x, y) ≤ C (x+ y, 0) = 0.Furthermore, given x, y ∈ [0, 1] such that x + y > 1, we have (x, y) ≺ (1, x+ y − 1),from whichC (x, y) ≤ C (1, x+ y − 1) = x+ y − 1.Then, for all x, y ∈ [0, 1]C (x, y) ≤ max (x+ y − 1, 0) =W (x, y) ,but, from the Fre´chet–Hoeffding bounds inequalities (1.13) it follows that C =W .Notice thatW is also the only Schur–constant (semi–)copula, as showed in Propo-sition 2.2.2.Now, we give some results about the class CSC of Schur–concave copulas.Proposition 10.1.5. The class CSC is a compact subset of C with respect to thetopology of uniform convergence.Proof. It is known that C is compact space with respect to the topology of uniformconvergence. But, if (Cn)n∈N is a sequence in CSC , then the pointwise limitC (x, y) = limn→+∞Cn (x, y)is Schur–concave. It follows that the set CSC is a closed subset of C, and therefore itis also compact.Proposition 10.1.6. The class CSC is a convex subset of C.Proof. Let (x1, x2) and (y1, y2) be two points in [0, 1]2 such that (x1, x2) ≺ (y1, y2)and suppose that C1 and C2 are Schur–concave copulas. Then, for every λ ∈ [0, 1]C(x1, x2) = λC1(x1, x2) + (1− λ)C2(x2, y2)≥ λC1(y1, y2) + (1− λ)C2(y1, y2) = C(y1, y2),which concludes the proof.Proposition 10.1.7. A copula C is Schur–concave if, and only if, the survival copulaCˆ associated with C is Schur–concave.Proof. If C is Schur–concave, then, given (x1, x2), (y1, y2) two points in ∆+ such that(x1, x2) ≺ (y1, y2), we have(1− x1, 1− x2) ≺ (1− y1, 1− y2),138 Chap. 10 Copulas and Schur–concavityfrom whichC(1− x1, 1− x2) ≥ C(1− y1, 1− y2),andx1 + x2 − 1 + C(1− x1, 1− x2) ≥ y1 + y2 − 1 + C(1− y1, 1− y2),Then Cˆ is Schur-concave. The same argument applies if Cˆ is assumed to be Schur–concaveIn view of Sklar’s Theorem, given a copula C and two univariate d.f.’s F and G, itis possible to construct a bivariate d.f. H(x, y) := C(F (x), G(y)) for every (x, y) ∈ R2.Now, it is useful to stress the fact that, with suitable marginal d.f.’s, Schur–concavecopulas may yield Schur–concave, –convex or constant bivariate d.f.’s (see [115]).10.2 Families of Schur–concave copulasTheorem 10.2.1. Every associative copula is Schur–concave.In order to prove this result, first we establish the following two lemmas.Lemma 10.2.1. An ordinal sum of Schur–concave copulas is a Schur–concave copula.Proof. Let {Ji = [ai, bi]}i∈I be a partition of the unit square and let {Ci}i∈I be afamily of Schur–concave copulas. Let C be the ordinal sum of {Ci}i∈I with respectto {Ji}i∈I, viz.C(x, y) :=ai + (bi − ai) Ci(x− aibi − ai ,y − aibi − ai), if (x, y) ∈ J2i ;M(x, y), otherwise.Notice that C is symmetric and we shall show that, if every Ci is Schur–concave, thenC is Schur–concave. Let (x1, x2), (y1, y2) be two points in ∆+ such that (x1, x2) ≺(y1, y2). Suppose that there exists an index i0 ∈ I such that (x1, x2), (y1, y2) ∈ J2i0 .We observe that (x1 − ai0bi0 − ai0,x2 − ai0bi0 − ai0)≺(y1 − ai0bi0 − ai0,y2 − ai0bi0 − ai0),that impliesCi0(x1 − ai0bi0 − ai0,x2 − ai0bi0 − ai0)≥ Ci0(y1 − ai0bi0 − ai0,y2 − ai0bi0 − ai0),since Ci0 is Schur–concave, and it follows C(x1, x2) ≥ C(y1, y2). Similarly, if (x1, x2)and (y1, y2) does not belong to J2i for all i ∈ I, since M is also Schur–concave, itfollows C(x1, x2) ≥ C(y1, y2). Finally, suppose that exists an index i0 such that(x1, x2) ∈ J2i0 and (y1, y2) 6∈ J2i for all i ∈ I. We set k := x1 + x2 = y1 + y2 and wedistinguish two cases.Chap. 10 Copulas and Schur–concavity 139Case 1. If 2ai0 ≤ k ≤ ai0 + bi0 , then (x1, x2) ≺ (k − ai0 , ai0) andC(x1, x2) ≥ C(k − ai0 , ai0) = ai0 ≥M(y1, y2) = C(y1, y2);hence C is Schur–concave.Case 2. If ai0 + bi0 < k < 2bi0 , then (x1, x2) ≺ (bi0 , k − bi0) andC(x1, x2) ≥ C(bi0 , k − bi0) = k − bi0 ≥M(y1, y2) = C(y1, y2),from which it follows that C is Schur–concave.Lemma 10.2.2. Every Archimedean copula is Schur–concave.Proof. Let (x1, x2) and (y1, y2) two points in [0, 1]2 such that (x1, x2) ≺ (y1, y2). Itfollows from Corollary 1.2.1 that there exists α ∈ [0, 1] such that, if α := 1− α, thenx1 = α y1 + α y2, x2 = α y1 + α y2.Let Cϕ be an Archimedean copula with additive generator ϕ. Since ϕ is convex andstrictly decreasingC(x1, x2) = C(α y1 + α y2, αy1 + α y2)= ϕ[−1] (ϕ(α y1 + α y2) + ϕ(α y1 + α y2))≥ ϕ[−1] (αϕ(y1) + αϕ(y2) + αϕ(y1) + αϕ(y2))= ϕ[−1] (ϕ(y1) + ϕ(y2)) = C(y1, y2),which concludes the proof.Proof. (Theorem 10.2.1) It was shown that M and every Archimedean copula areSchur–concave, moreover the ordinal sum of two Schur–concave copulas is Schur–concave too. In view of Representation Theorem for associative copulas (Theorem1.6.9), the assertion follows.Here we give some other examples of Schur–concave copulas.Example 10.2.1 (The Fre´chet family). Every copula Cα,β belonging to the Fre´chetfamily (see Example 1.6.2), defined byCα,β(x, y) = αM(x, y) + (1− α− β)Π(x, y) + βW (x, y)is Schur–concave, because it is a convex sum of Schur–concave copulas.Example 10.2.2 (The FGM family). For all x, y ∈ [0, 1] and θ ∈ [−1, 1]Cθ(x, y) = xy + θxy (1− x) (1− y)140 Chap. 10 Copulas and Schur–concavityis a member of the FGM family (see Example 1.6.3). For every x, y ∈ [0, 1] we have∂1Cθ(x, y) = y + θy (1− x) (1− y)− θxy (1− y),∂2Cθ(x, y) = x+ θx (1− x) (1− y)− θxy (1− x).As a consequence of the inequality |1− x− y + 2xy| ≤ 1, which holds for all x and yin [0, 1], if x ≥ y we have∂2Cθ(x, y)− ∂1Cθ(x, y) = (x− y)[1 + θ(1− x− y + 2xy)] ≥ 0.Thus, it follows from Proposition 10.1.3 that Cθ is Schur–concave.Example 10.2.3 (The Plackett family). For all u, v ∈ [0, 1] and θ > 0, θ 6= 1,Cθ(u, v) =[1 + (θ − 1) (u+ v)]−√[1 + (θ − 1) (u+ v)]2 − 4θuv (θ − 1)2 (θ − 1)is a family of copulas, known as Plackett family (see [130]). For all x, y ∈ [0, 1], wehave∂1Cθ(u, v) =12− 1 + (θ − 1) (u+ v) − 2θv2√[1 + (θ − 1) (u+ v)]2 − 4θuv (θ − 1) ,∂2Cθ(u, v) =12− 1 + (θ − 1) (u+ v) − 2θu2√[1 + (θ − 1) (u+ v)]2 − 4θuv (θ − 1) .Moreover, for u ≥ v, it follows that∂2Cθ(u, v)− ∂1Cθ(u, v) = θ (u− v)2√[1 + (θ − 1) (u+ v)]2 − 4θuv (θ − 1) ≥ 0.Thus Cθ is Schur–concave.10.3 Solution of an open problem for associativecopulasRecently, E.P. Klement, R. Mesiar and E. Pap ([85]) posed some open problemsconcerning triangular norms and related operators. In particular, the following prob-lem was formulated:Problem 10.3.1. Let T be a continuous Archimedean t–norm. Prove or disprovethat:T (max{x− a, 0},min{x+ a, 1}) ≤ T (x, x) (10.1)holds for all x ∈ [0, 1] and for all a ∈ ]0, 1/2[.Chap. 10 Copulas and Schur–concavity 141In particular, the authors added that “a positive solution of this problem wouldinduce a new characterization of associative copulas”. This comment spurs us toinvestigate inequality (10.1) in the class of copulas: to this end, the notion of Schur–concavity will be useful.First, notice that inequality (10.1) is not true for every copula.Example 10.3.1. Let C be the copula given in [114, Example 3.3],C(x, y) :=x, if 0 ≤ x ≤ y2≤ 12;y2, if 0 ≤ y2< x < 1− y2;x+ y − 1, if 12≤ 1− y2≤ x ≤ 1.ThenC(410,610)=310> C(12,12)=14.Note that C is not associative:C(C(12,12),12)= C(14,12)=146= 18= C(12, C(12,12)).But, in general, we haveLemma 10.3.1. Let A be a semicopula. If A is Schur–concave, then A satisfies(10.1).Proof. Let a be in ]0, 1/2[. We distinguish three cases. If x ≤ a, then (10.1) followssince A is positive. If a < x ≤ 1 − a, then (10.1) is equivalent to A(x − a, x + a) ≤A(x, x), which is a direct consequence of the Schur–concavity. If x > 1 − a, then(10.1) is equivalent to x − a ≤ A(x, x) and this last inequality follows from the factthatA(x, x) ≥ A(2x− 1, 1) = 2x− 1 > x− a.Lemma 10.3.1 and Theorem 10.2.1 together yield:Theorem 10.3.1. If C is an associative copula, then C satisfies (10.1).Notice that, if a copula C satisfies (10.1), then it need not be associative.Example 10.3.2. We consider the FGM family of copulas given, for all x, y ∈ [0, 1]and θ ∈ [−1, 1], by Cθ(x, y) = xy + θxy (1 − x) (1 − y). From Example 10.2.2, Cθ isSchur–concave, and thus satisfies (10.1), but, if θ 6= 0, Cθ is not associative.Notice also that, if a copula C satisfies (10.1), then it need not be Schur–concave.142 Chap. 10 Copulas and Schur–concavityExample 10.3.3. Let C be the copula defined byC(x, y) :=13M(3x, 3y − 2), if (x, y) ∈[0, 13]× [ 23 , 1] ;13M(3x− 1, 3y − 1), if (x, y) ∈[13 ,23]× [ 13 , 23] ;13M(3x− 2, 3y), if (x, y) ∈[23 , 1]× [0, 13] ;W (x, y), otherwise.This copula is obtained by using the block–based construction method introducedin [28]. Simple, but tedious, calculations show that C satisfies (10.1), but C is notSchur–concave. In fact, given the points (2/10, 7/10) and (3/10, 6/10), we haveC(310,610)= 0 <130= C(210,710),which implies that C is not Schur–concave.Remark 10.3.1. A geometrical interpretation can be given of the difference betweeninequality (10.1) and Schur–concavity. If z = C(s, t) is the surface associated witha copula C that satisfies (10.1), the intersections of the surface with all the verticalplanes of the form s+ t = 2x, for all x ∈ [0, 1] and s ∈ [0, x], are curves that take themaximum value in the point (x, x). But, if C is Schur–concave, we have the strongercondition that such curves are also decreasing from (x, x) to (2x, 0) (resp. (2x−1, 1)).10.3.1 Discussion in the class of triangular normsIn the class of continuous Archimedean t–norms, inequality (10.1) was character-ized in [67] (see also [98, 127]).Theorem 10.3.2. Let T be a continuous Archimedean t–norm with additive generatort. Let ξ be defined by ξ := t−1 (t(0)/2). Then T satisfies (10.1), for all a ∈ ]0, 1/2[and x ∈ [0, 1], if, and only if, the two following statements hold:(a) for all z ∈ ]0,min{ξ, 1− ξ}[, t(ξ − z) + t(ξ + z) ≥ 1;(b) t is convex on [ξ, 1].In particular, if T is strict (viz. t(0) = +∞), then the following statements are equiv-alent:(a’) T satisfies (10.1), for all a ∈ ]0, 1/2[ and x ∈ [0, 1];(b’) t is convex on [0, 1].On the other hand, we have also the characterization of continuous Archimedeant–norms that are Schur–concave (see [1]).Theorem 10.3.3. Let T be a continuous Archimedean t–norm with additive generatort. Then we have:Chap. 10 Copulas and Schur–concavity 143(a) if T is strict, then T is Schur–concave if, and only if, t is convex;(b) if T is nilpotent, then T is Schur–concave if, and only if, t satisfies the followinginequality:t (αx+ (1− α)y) + t ((1− α)x+ αy) ≤ t(x) + t(y)for every α in [0, 1] and for all x, y in [0, 1] such that t(x) + t(y) ≤ 1.From the two previous results, we deriveTheorem 10.3.4. Let T be a strict Archimedean t–norm with additive generator t.The following statements are equivalent:(i) T is a copula;(ii) T is Schur–concave;(iii) T satisfies (10.1).Proof. From Theorem 1.6.6, T is a copula if, and only if, the additive generator tis convex and, then, T is Schur–concave (Theorem 10.3.3). Moreover, from Lemma10.3.1, (ii) implies (iii), which, in its turn, is equivalent to the convexity of t (Theorem10.3.2), which concludes the proof.Remark 10.3.2. The previous result also holds in the case of a continuous t–normT which is jointly strictly monotone, i.e. T (x, y) < T (x, z) whenever x > 0 and y < z(see [88]).Looking at Theorem 10.3.4 in the class of nilpotent t–norm, we have (i) =⇒(ii) =⇒ (iii). But, there exists a Schur–concave nilpotent t–norm T , which is nota copula: consider, for example, a t–norm additively generated by t(x) := 1+cos(pix)2(see [1, Example 2.1]). Moreover, in the class of nilpotent t–norms, inequality (10.1)does not imply Schur–concavity as the following example shows.Example 10.3.4. Consider a t–norm T with additive generator t given byt(x) :=1− x10 , if x ∈[0, 110];− 49√210(9√2−10)(x− 110)+ 99100 if x ∈]110 , 1− 1√2];(1− x)2, otherwise.Then T satisfies the assumptions of Theorem 10.3.2, and thus the inequality (10.1),butT (5/100, 95/100) = 25/1000 > 0 = T (1/10, 9/10),which implies that T is not Schur–concave.

Copulas and Schur–concavity

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Copulas and Schur–concavity

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