Multivariate t and beta distributions associated with the multivariate F distribution

Abstract.

Relationships between F, skew t and beta distributions in the univariate case are in this paper extended in a natural way to the multivariate case. The result is two new distributions: a multivariate t/skew t distribution (on ℜ^m) and a multivariate beta distribution (on (0,1)^m). A special case of the former distribution is a new multivariate symmetric t distribution. The new distributions have a natural relationship to the standard multivariate F distribution (on (ℜ⁺)^m) and many of their properties run in parallel. We look at: joint distributions, mathematically and graphically; marginal and conditional distributions; moments; correlations; local dependence; and some limiting cases.