Non-parametric estimator of a multivariate madogram for missing-data and extreme value framework

TL;DR

This paper develops a nonparametric estimator for the multivariate madogram in the context of missing data and extreme value analysis, providing theoretical properties and simulation validation.

Contribution

It introduces a new nonparametric estimator for the multivariate madogram that accounts for missing data, with proven asymptotic properties and explicit variance formulas.

Findings

Asymptotic normality of the estimator is established.

Explicit covariance and variance formulas are derived.

Simulation studies confirm the estimator's performance.

Abstract

The modeling of dependence between maxima is an important subject in several applications in risk analysis. To this aim, the extreme value copula function, characterised via the madogram, can be used as a margin-free description of the dependence structure. From a practical point of view, the family of extreme value distributions is very rich and arise naturally as the limiting distribution of properly normalised component-wise maxima. In this paper, we investigate the nonparametric estimation of the madogram where data are completely missing at random. We provide the functional central limit theorem for the considered multivariate madrogram correctly normalized, towards a tight Gaussian process for which the covariance function depends on the probabilities of missing. Explicit formula for the asymptotic variance is also given. Our results are illustrated in a finite sample setting with a simulation study.