Comportement extrémal des copules diagonales et de Bertino

Abstract

The maximal attractors of bivariate diagonal and Bertino copulas are determined under suitable regularity conditions. Some consequences of these facts are drawn, namely bounds on the maximal attractor of a symmetric copula with a given diagonal section, and bounds on Spearman’s rho and Kendall’s tau for an exchangeable extreme-value copula whose upper-tail dependence coefficient is known. Some of these results are then extended to the case of arbitrary bivariate copulas and to multivariate copulas. Classification Mathématique (2020). 60G70, 62G32. Financement. Ce travail a bénéficié de l’appui financier du Secrétariat des Chaires de recherche du Canada, du Conseil de recherches en sciences naturelles et en génie du Canada, ainsi que de l’Institut Trottier pour la science et la politique publique. Manuscrit reçu le 9 juin 2020, révisé le 15 octobre 2020, accepté le 21 octobre 2020. Abridged English version A copula C is the distribution function of a random vector (U1, . . . ,Uk ) with uniform margins on the unit interval. Its diagonal section ∆(C ) is the distribution of max(U1, . . . ,Uk ). Several authors have considered the question of what can be said about C when ∆(C ) is known. In dimension k = 2, point-wise lower and upper bounds on the joint distribution C of a random pair (U ,V ) of exchangeable uniform random variables are given by the Fréchet–Hoeffding copulas. The latter correspond to the cases of comonotonic dependence in which either V = 1−U or V =U almost ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/ 1158 Christian Genest et Magid Sabbagh surely. Nelsen et al. [26] showed that when∆(C ) = δ is known and C is symmetric, it is possible to tighten these bounds. Specifically, one has Bδ(u, v) ≤C (u, v) ≤ Kδ(u, v), for all (u, v) ∈ [0,1], where Bδ(u, v) = (u ∧ v)− inf{t −δ(t ) : t ∈ [u ∧ v,u ∨ v]}, defines the Bertino copula [2] with diagonal section δ and Kδ(u, v) = u ∧ v ∧ {δ(u)+δ(v)}/2 is another copula with diagonal section δ which Fredricks and Nelsen [8] called a “diagonal copula.” Here and below, a ∧b = min(a,b) and a ∨b = max(a,b) for any reals a and b. In this paper, the extremal behavior of the copulas Bδ and Kδ is determined under suitable regularity assumptions on δ. It is first shown in Section 2 that if δ admits a left-sided derivative δ′ at 1, say d = δ′(1−), then Kδ belongs to the max domain of attraction of the copula with parameter θ = d/2 ∈ [1/2,1] defined, for all (u, v) ∈ [0,1]2, by Dθ(u, v) = u ∧ v ∧ (uv) . Further assume that there exits a real 2 ∈ (0,1) such that the map δ̂ : [0,1] → [0,1] defined at each t ∈ [0,1] by δ̂(t ) = t−δ(t ) is decreasing on the interval (2,1). Under this additional condition, it is shown in Section 3 that Bδ belongs to the max domain of attraction of the Cuadras–Augé copula with parameter θ = 2−d ∈ [0,1] defined, for all (u, v) ∈ [0,1]2, by Qθ(u, v) = (uv)1−θ(u ∧ v) . Various consequences of these results are mentioned in Section 4. First and foremost, if C is a symmetric bivariate copula with diagonal section δ meeting the above requirements, and if C∗ denotes its max attractor, which is assumed to exist, then for all (u, v) ∈ [0,1]2, Q2−d (u, v) ≤C∗(u, v) ≤ Dd/2(u, v). This string of inequalities immediately entails that the upper-tail dependence coefficient associated with C is given by Λ(C ) =Λ(C∗) =Λ(Q2−d ) =Λ(Dd/2) = 2−d . Moreover, if C∗ is symmetric with upper-tail dependence coefficientΛ(C∗) =λ, say, and if ρ(C∗) and τ(C∗) respectively denote the values of Spearman’s rho and Kendall’s tau associated with C∗, then 3λ/(4−λ) ≤ ρ(C∗) ≤ 3λ(8−5λ)/(4−λ)2 and λ/(2−λ) ≤ τ(C∗) ≤λ. These bounds settle a question raised in [22], which Jaworski [18] recently solved differently. Proposition 4 in Section 5 then shows how the symmetry assumption on C can be relaxed. Finally, Section 6 comments briefly on possible extensions to arbitrary dimension k > 2. It is pointed out there that the search for a lower bound is hindered by the fact that the k-variate extension of the bivariate Bertino copula is generally not a distribution unless the diagonal section is Lipschitz increasing of degree k/(k−1), as reported by Arias-García et al. [1]. In contrast, the k-variate extension of the diagonal copula introduced by Jaworski [17] does have a max attractor under the same assumptions as in the bivariate case. See Proposition 5 for details.