Estimation of the Spectral Measure from ConvexCombinations of Regularly Varying RandomVectors

TL;DR

This paper introduces new estimators for the spectral measure of regularly varying vectors, enabling better inference of extremal dependence structures through convex combinations, with proven asymptotic properties and practical validation.

Contribution

It develops novel estimators for spectral measure characteristics based on convex combinations, with asymptotic normality and minimal variance demonstrated.

Findings

Asymptotic normality of estimators established

Minimal asymptotic variance achieved via subsampling bootstrap

Validated on simulated and real data

Abstract

The extremal dependence structure of a regularly varying random vector Xis fully described by its limiting spectral measure. In this paper, we investigate how torecover characteristics of the measure, such as extremal coefficients, from the extremalbehaviour of convex combinations of components of X. Our considerations result in aclass of new estimators of moments of the corresponding combinations for the spectralvector. We show asymptotic normality by means of a functional limit theorem and, focusingon the estimation of extremal coefficients, we verify that the minimal asymptoticvariance can be achieved by a plug-in estimator using subsampling bootstrap. We illustratethe benefits of our approach on simulated and real data.