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

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.

Sample from copula: a COPPY module

“…We now proceed to a case study where we use our Python module to assess, under a finite sample framework, the asymptotic properties of an estimator of the λ-madogram when data are completely missing at random (MCAR). This case study comes from numerical results of [Boulin et al, 2021].…”

“…We denote by p the probability of observing completely a realization from X, that is p = P(I 0 = 1, I 1 = 1). In [Boulin et al, 2021], hybrid and corrected estimators, respectively denoted as νH n and νH * n , are proposed to estimate nonparametrically the λ-madogram in presence of missing data completely at random. Furthermore, a closed expression of their asymptotic variances for λ ∈]0, 1[ is also given.…”

“…where ν(λ) is defined in (10), S H (p 0 , p 1 , p, λ) and S H * (p 0 , p 1 , p, λ) are the asymptotic variances of the random variables where the closed expression is given in [Boulin et al, 2021], Proposition 1.…”