F-t joint distributions on real Siegel domains

Abstract In statistics, the homogeneous cones are considered as the general parameter spaces for variance matrices related with the graphical models. For the argument of probability distributions, the Wishart distributions are defined based on the analysis of homogeneous cones, as the image of the normal distributions by quadratic maps. Now, the parameter space of pairs of variance matrix and mean vector is identified with more general homogeneous domain, called the real Siegel domain. In the present work, we consider random variables valued in the real Siegel domain associated with the quadratic map of homogeneous cone, and derive their probability density functions via the transformation group. This derivation is also based on the analysis on homogeneous cones including the gamma and beta functions due to Gindikin. We call our result the F - t joint distribution on the real Siegel domain, since its marginal distributions give an extension of the F and t -distributions in 1-dimensional case. Especially, it enables us to describe a joint estimation and a simultaneous hypothesis testing for the pair of parameters of normal distribution.