Families of Multivariate Distributions

TLDR

Interest in bivariate distributions with marginal parameters has long motivated unified derivations, and mixture models are recognized as useful tools in this context. This study uses mixture models to derive new families of bivariate distributions with marginal parameters and to suggest multivariate extensions. The mixture models are treated as multivariate proportional‑hazards models with random proportionality constants, revealing transparent dependence properties and enabling simulation of data. The models expose positive‑dependency properties, provide a simulation method, and generate several new bivariate families—some extending known distributions with an extra parameter and others that appear entirely novel.

Abstract

Abstract For many years there has been an interest in families of bivariate distributions with marginals as parameters. Genest and MacKay (1986a,b) showed that several such families that appear in the literature can be derived by a unified method. A similar conclusion is obtained in this article through the use of mixture models. These models might be regarded as multivariate proportional hazards models with random constants of proportionality. The mixture models are useful for two purposes. First, they make some properties of the derived distributions more transparent; the positive-dependency property of association is sometimes exposed, and a method for simulation of data from the distributions is suggested. But the mixture models also allow derivation of several new families of bivariate distributions with marginals as parameters, and they indicate obvious multivariate extensions. Some of the new families of bivariate distributions given in this article extend known distributions by adding a parameter to make them more flexible. Other families are derived that appear to be entirely new.