Distortion representations of multivariate distributions

The univariate distorted distributions 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 and proportional reversed hazard rate 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 representation, especially when the marginal distributions may not be easily handled. Several 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.