Higher order tail densities of copulas and hidden regular variation

A notion of higher order tail densities for copulas is introduced using multivariate regular variation of copula densities, and densities of multivariate extremes with various margins can then be studied in a unified fashion. We show that the tail of a multivariate density can be decomposed into the tail density of the underlying copula, coupled with marginal tail transforms of the three types: Fréchet, Gumbel, and Weibull types. We also derive the relation between the tail density and tail order functions of a copula in the context of hidden regular variation. Some illustrative examples are given.