The multivariate Piecing-Together approach revisited

TL;DR

This paper revisits the multivariate Piecing-Together approach, providing an exact tail representation, a variant with empirical copulas, and extending it to a functional version, enhancing understanding and applicability.

Contribution

It offers a precise tail characterization, introduces an empirical copula variant, and develops a functional extension of the multivariate PT approach.

Findings

Exact representation of the PT-copula's upper tail.

Introduction of a variant based on empirical copula.

Development of a functional PT version.

Abstract

The univariate Piecing-Together approach (PT) fits a univariate generalized Pareto distribution (GPD) to the upper tail of a given distribution function in a continuous manner. A multivariate extension was established by Aulbach et al. (2012a): The upper tail of a given copula C is cut off and replaced by a multivariate GPD-copula in a continuous manner, yielding a new copula called a PT-copula. Then each margin of this PT-copula is transformed by a given univariate distribution function. This provides a multivariate distribution function with prescribed margins, whose copula is a GPD-copula that coincides in its central part with C. In addition to Aulbach et al. (2012a), we achieve in the present paper an exact representation of the PT-copula's upper tail, giving further insight into the multivariate PT approach. A variant based on the empirical copula is also added. Furthermore our findings enable us to establish a functional PT version as well.