Choice of smoothing parameter in multivariate copula-based tail coefficients

Abstract

Multivariate tail coefficients are of interest when one is trying to summarize dependence of extremes. The difficulty is that these coefficients are defined as limits and thus to estimate them from data one needs to choose a kind of smoothing parameter kn. A standard approach in such problems is to choose the smoothing parameter as a minimum of an estimated asymptotic mean squared error (AMSE). In this paper several ways of estimating AMSE are suggested and investigated. We focus on estimating (Frahm’s) extremal dependence coefficient. In order to derive AMSE of this estimator we provide its asymptotic representation. Two methods (fully or partially nonparametric) of estimating AMSE are suggested. An extensive simulation study shows that both suggested methods usually slightly outperform the available alternatives with the fully nonparametric method being recommended. Finally the different ways of choosing kn are illustrated on a real data set.