Families of multivariate distributions involving the Rosenblatt construction

Recently, Jones pointed out a useful device for enriching families of univariate distributions. Typically, one may construct a random variable with distribution F by considering F-1 (U), where U is a uniform(0, 1) random variable. Jones suggested replacing the uniform random variable by a beta-distributed random variable. In this article an analogous enriching process is applied to the Rosenblatt construction of multivariate distributions. Several parametric families are introduced using this construction, together with discussion of appropriate parameter estimation strategies. It turns out that the resulting families of distributions are very flexible and easy to estimate. The method is illustrated with simulated and real data.