On estimating extremal dependence structures by parametric spectral measures

TL;DR

This paper introduces a parametric spectral measure approach for estimating extremal dependence structures in multivariate extreme value analysis, especially useful with sparse data, and provides theoretical and simulation validation.

Contribution

It proposes a novel parametric family of spectral measures based on convex hulls and develops asymptotic theory for parameter estimation and dependence structure inference.

Findings

Asymptotic distributions for estimators are derived.

Simulation studies demonstrate finite sample performance.

The method effectively captures a wide range of extremal dependence structures.

Abstract

Estimation of extreme value copulas is often required in situations where available data are sparse. Parametric methods may then be the preferred approach. A possible way of defining parametric families that are simple and, at the same time, cover a large variety of multivariate extremal dependence structures is to build models based on spectral measures. This approach is considered here. Parametric families of spectral measures are defined as convex hulls of suitable basis elements, and parameters are estimated by projecting an initial nonparametric estimator on these finite-dimensional spaces. Asymptotic distributions are derived for the estimated parameters and the resulting estimates of the spectral measure and the extreme value copula. Finite sample properties are illustrated by a simulation study.