Nonparametric estimation of the spectral measure of an extreme value distribution

Abstract

Let 1 1 n n be a random sample from a bivariate distribution function F in the domain of max-attraction of a distribution function G. This G is characterised by the two extreme value indices and its spectral or angular measure. The extreme value indices determine both the marginals and the spectral measure determines the dependence structure of G. One of the main issues in multivariate extreme value theory is the estimation of this spectral measure. We construct a truly nonparametric estimator of the spectral measure, based on the ranks of the above data. Under natural conditions we prove consistency and asymptotic normality for the estimator. In particular, the result is valid for all values of the extreme value indices. The theory of (local) empirical processes is indispensable here. The results are illustrated by an application to real data and a small simulation study.

Bayesian Inference with M-splines on Spectral Measure of Bivariate Extremes

“…In this section, we compare the performances of the spectral measure Bayes-M-splines estimator introduced in this paper with three estimators proposed recently in the literature in Einmahl et al (2001), Einmahl and Segers (2009), and de Carvalho et al (2013). To be complete in this section, let us recall briefly these three methods with which we aim to compare ours.…”

“…-Empirical Spectral Measure (ESM): The authors in Einmahl et al (2001) proposed from the previous transformation an empirical estimatorĤ ESM of the spectral measure defined byĤ…”

“…The literature on spectral measure usually focuses on frequentist inference within parametric approaches; see for instance (Coles andTawn 1991, 1994;Joe et al 1992;Smith 1994;Ledford and Tawn 1996;de Haan et al 2008;Boldi and Davison 2007) and nonparametric procedures; see for instance (de Haan and de Rond 1998;de Haan and Sinha 1999;Einmahl et al 2006;Einmahl et al 2001;Schmidt and Stadtmüller 2006;Einmahl and Segers 2009). Recently, using the method of Lagrange multipliers, a constrained piecewise linear estimate has been proposed in Einmahl and Segers (2009) in order to incorporate the moment constraint characterizing the class of spectral measure.…”

Testing the Multivariate Regular Variation Model

“…We start by proving (A.1) without the weight function x 1/2−ζ . This is achieved by applying Lemma 3.1 in Einmahl, de Haan, and Sinha (1997). Write U i = 1 − F R (R i ), then U 1 , .…”

“…In order to apply Lemma 3.1 in Einmahl, de Haan, and Sinha (1997), we only need to check that as n → ∞,…”