Projection estimators of Pickands dependence functions

Abstract

Abstract:The authors consider the construction of intrinsic estimators for the Pickands dependence function of an extreme-value copula. They show how an arbitrary initial estimator can be modified to satisfy the required shape constraints. Their solution consists in projecting this estimator in the space of Pickands functions, which forms a closed and convex subset of a Hilbert space. As the solution is not explicit, they replace this functional parameter space by a sieve of finite-dimensional subsets. They establish the asymptotic distribution of the projection estimator and its finite-dimensional approximations, from which they conclude that the projected estimator is at least as efficient as the initial one. Estimation par projection de la fonction de dépendance de PickandsRésumé : Les auteurs s'intéressentà la construction d'estimateurs intrinsèques de la fonction de dépendan-ce de Pickands d'une copule des valeurs extrêmes. Ils montrent comment un estimateur initial quelconque peutêtre modifié pour satisfaire les contraintes de forme voulues. Leur solution consisteà projeter cet estimateur dans l'espace des fonctions de Pickands, qui forme un sous-ensemble convexe fermé d'un espace de Hilbert. Comme la solution n'est pas explicite, ils remplacent cet espace paramétrique fonctionnel par une succession d'approximations de dimension finie. Ilsétablissent la distribution asymptotique de la projection de l'estimateur et de ses approximations de dimension finie, ce qui leur permet de conclure que l'estimateur projeté est au moins aussi efficace que l'estimateur initial.

Using B-splines for nonparametric inference on bivariate extreme-value copulas

“…In the pilot study, values of M = 50 and 100 produced satisfactory results for samples of size 200 and 500, respectively; higher values of M provided no detectable improvement in performance. The study also confirmed that P and CFG always have smaller MSE and MISE than the non-intrinsic estimators from which they were built, as stated in Fils-Villetard et al (2008). However, the improvement provided by the intrinsic versions over their non-intrinsic counterparts appeared to be negligible in all scenarios considered.…”

“…These intrinsic estimators, derived by the projection method of Fils-Villetard et al (2008), are denoted P and CFG , respectively. As explained by these authors, the computation of the projection involves an infinite-dimensional minimization problem under constraints for which no explicit solution exists.…”

“…Finally, the performance of P and CFG depends on the value of M. Fils-Villetard et al (2008) state that the latter should be taken as large as feasible, given computing time and power. In the pilot study, values of M = 50 and 100 produced satisfactory results for samples of size 200 and 500, respectively; higher values of M provided no detectable improvement in performance.…”

“…Comparisons are made with the rank-based versions of the Pickands and Capéraà-Fougères-Genest (CFG) estimators introduced in Genest and Segers (2009). In order to ensure fair comparisons, the latter estimators were first made intrinsic via the projection method described in Fils-Villetard et al (2008). In the cases considered, the new estimator was generally better than the Pickands estimator and a close competitor to the CFG estimator, especially at moderate levels of dependence.…”

Local robust estimation of the Pickands dependence function

“…Statistical inference on the bivariate function C is therefore equivalent to the statistical inference on the one-dimensional function A. Estimating this function A has been extensively studied in the literature. We can mention, among others, Capéraà, Fougères and Genest (1997), Fils-Villetard, Guillou and Segers (2008) or Bücher, Dette and Volgushev (2011).…”