General Multivariate Dependence using Associated Copulas

Abstract

This paper studies the general multivariate dependence and tail dependence of a random vector. We analyse the dependence of variables going up or down, covering the 2 d orthants of dimension d and accounting for non-positive dependence. We extend definitions and results from positive to general dependence using the associated copulas. We study several properties of these copulas and present general versions of the tail dependence functions and tail dependence coefficients. We analyse the perfect dependence models, elliptical copulas and Archimedean copulas. We introduce the monotonic copulas and prove that the multivariate Student’s t copula accounts for all types of tail dependence simultaneously while Archimedean copulas with strict generators can only account for positive tail dependence.