Non‐parametric Estimation of Tail Dependence

TLDR

Dependencies between extreme events and the concept of tail dependence are increasingly important in risk management, with multivariate extreme‑value theory providing the natural framework for modeling such extremal dependence. The paper embeds tail dependence into tail copulae to describe tail dependence more generally. The authors develop non‑parametric estimators for tail copulae and tail dependence, proving weak convergence, asymptotic normality, and strong consistency via a functional delta method, and study weak convergence of upper‑order rank statistics for extreme events. A simulation study compares the introduced estimators, and two financial data sets are analyzed using the methods. Abstract.

Abstract

Abstract. Dependencies between extreme events (extremal dependencies) are attracting an increasing attention in modern risk management. In practice, the concept of tail dependence represents the current standard to describe the amount of extremal dependence. In theory, multi‐variate extreme‐value theory turns out to be the natural choice to model the latter dependencies. The present paper embeds tail dependence into the concept of tail copulae which describes the dependence structure in the tail of multivariate distributions but works more generally. Various non‐parametric estimators for tail copulae and tail dependence are discussed, and weak convergence, asymptotic normality, and strong consistency of these estimators are shown by means of a functional delta method. Further, weak convergence of a general upper‐order rank‐statistics for extreme events is investigated and the relationship to tail dependence is provided. A simulation study compares the introduced estimators and two financial data sets were analysed by our methods.