Maximum empirical likelihood estimation of the spectral measure of an extreme-value distribution

TL;DR

This paper introduces a nonparametric maximum empirical likelihood estimator for the spectral measure in multivariate extreme-value distributions, ensuring moment constraints and demonstrating improved performance through theory and simulations.

Contribution

It proposes a novel empirical likelihood estimator for the spectral measure that guarantees moment constraints, with proven asymptotic normality and practical applicability.

Findings

Estimator satisfies spectral measure constraints

Asymptotic normality established under tail independence

Simulation shows improved estimator performance

Abstract

Consider a random sample from a bivariate distribution function $F$ in the max-domain of attraction of an extreme-value distribution function $G$. This $G$ is characterized by two extreme-value indices and a spectral measure, the latter determining the tail dependence structure of $F$. A major issue in multivariate extreme-value theory is the estimation of the spectral measure $\Phi_p$ with respect to the $L_p$ norm. For every $p\in[1,\infty]$, a nonparametric maximum empirical likelihood estimator is proposed for $\Phi_p$. The main novelty is that these estimators are guaranteed to satisfy the moment constraints by which spectral measures are characterized. Asymptotic normality of the estimators is proved under conditions that allow for tail independence. Moreover, the conditions are easily verifiable as we demonstrate through a number of theoretical examples. A simulation study shows a substantially improved performance of the new estimators. Two case studies illustrate how to implement the methods in practice.