# Tail Densities of Skew-Elliptical Distributions

## TL;DR

This paper derives explicit tail densities for skew-elliptical copulas, highlighting how skewness influences tail behavior under different tail conditions, with applications to skew-normal and skew-t distributions.

## Contribution

It provides a novel explicit form for tail densities of skew-elliptical copulas based on tail properties of the density generator and skewness parameters.

## Key findings

- Tail densities depend on tail properties of the generator.
- Skewness affects tail densities more in heavy tail cases.
- Explicit formulas for skew-normal and skew-t distributions.

## Abstract

Skew-elliptical distributions constitute a large class of multivariate distributions that account for both skewness and a variety of tail properties. This class has simpler representations in terms of densities rather than cumulative distribution functions, and the tail density approach has previously been developed to study tail properties when multivariate densities have more tractable forms. The special skew-elliptical structure allows for derivations of specific forms for the tail densities for those skew-elliptical copulas that admit probability density functions, under heavy and light tail conditions on density generators. The tail densities of skew-elliptical copulas are explicit and depend only on tail properties of the underlying density generator and conditions on the skewness parameters. In the heavy tail case skewness parameters affect tail densities of the skew-elliptical copulas more profoundly than that in the light tail case, whereas in the latter case the tail densities of skew-elliptical copulas are only proportional to the tail densities of symmetrical elliptical copulas. Various examples, including tail densities of skew-normal and skew-t distributions, are given.