Operator Tail Dependence of Copulas

TL;DR

This paper introduces a new operator-based tail dependence concept for copulas, expanding the understanding of multivariate tail behavior with diverse marginal scaling, and explores its implications for regularly varying distributions.

Contribution

It develops a novel operator tail dependence framework for copulas, generalizing standard tail dependence and linking it to multivariate regular variation with diverse margins.

Findings

Operator tail dependence encompasses standard tail dependence as a special case.

Copulas with operator tail dependence produce a broad class of multivariate regularly varying distributions.

Under mild conditions, the copula of such distributions exhibits standard tail dependence of order 1.

Abstract

A notion of tail dependence based on operator regular variation is introduced for copulas, and the standard tail dependence used in the copula literature is included as a special case. The non-standard tail dependence with marginal power scaling functions having possibly distinct tail indexes is investigated in detail. We show that the copulas with operator tail dependence, incorporated with regularly varying univariate margins, give rise to a rich class of the non-standard multivariate regularly varying distributions. We also show that under some mild conditions, the copula of a non-standard multivariate regularly varying distribution has the standard tail dependence of order 1. Some illustrative examples are given.