Distortion Representations of Multivariate Distributions

TL;DR

This paper extends univariate distorted distributions to multivariate cases, offering an alternative to copulas for modeling complex dependencies, with applications in various statistical models and dependent data analysis.

Contribution

It introduces multivariate distorted distributions, broadening their application scope and providing a new modeling approach when marginal distributions are difficult to handle.

Findings

Multivariate distortions can serve as an alternative to copulas.

Applications include dependent ordered data and joint lifetimes.

Examples demonstrate practical modeling benefits.

Abstract

The univariate distorted distribution were introduced in risk theory to represent changes (distortions) in the expected distributions of some risks. Later they were also applied to represent distributions of order statistics, coherent systems, proportional hazard rate (PHR) and proportional reversed hazard rate (PRHR) models, etc. In this paper we extend this concept to the multivariate setup. We show that, in some cases, they are a valid alternative to the copula representations especially when the marginal distributions may not be easily handled. Several relevant examples illustrate the applications of such representations in statistical modeling. They include the study of paired (dependent) ordered data, joint residual lifetimes, order statistics and coherent systems.