Nonparametric estimation of multivariate extreme-value copulas

TL;DR

This paper develops nonparametric estimators for multivariate extreme-value copulas based on the Pickands dependence function, ensuring shape constraints and demonstrating their theoretical properties and finite-sample performance.

Contribution

It introduces a new rank-based nonparametric estimation method for multivariate extreme-value copulas that enforces shape constraints via least-squares approximation.

Findings

Weak convergence of estimators established

Finite-sample performance demonstrated through simulations

Shape constraints effectively enforced in estimation

Abstract

Extreme-value copulas arise in the asymptotic theory for componentwise maxima of independent random samples. An extreme-value copula is determined by its Pickands dependence function, which is a function on the unit simplex subject to certain shape constraints that arise from an integral transform of an underlying measure called spectral measure. Multivariate extensions are provided of certain rank-based nonparametric estimators of the Pickands dependence function. The shape constraint that the estimator should itself be a Pickands dependence function is enforced by replacing an initial estimator by its best least-squares approximation in the set of Pickands dependence functions having a discrete spectral measure supported on a sufficiently fine grid. Weak convergence of the standardized estimators is demonstrated and the finite-sample performance of the estimators is investigated by means of a simulation experiment.