AN EXACT DECOMPOSITION THEOREM AND A UNIFIED VIEW OF SOME RELATED DISTRIBUTIONS FOR A CLASS OF EXPONENTIAL TRANSFORMATION MODELS ON SYMMETRICAL CONES

A class of exponential transformation models is defined on symmetric c ones OMEGA with the group of automorphisms on OMEGA as the acting grou p. We show that these models are reproductive and the exponent of thei r joint distribution for a given sample of size q can be split into q independent components, introducing one sample point at a time. The au tomorphism group can be factorized into the group of positive dilation and another group. Accordingly, the symmetric cone OMEGA can be seen as the direct product of R+ and a unit orbit, and every x in OMEGA can be identified by its orbital decomposition. We derive the distributio ns of the independent components of the exponent, of the ''length'' of x, the ''direction'' of x, the conditional distribution of the direct ion given the length and other distributions for a given sample. The W ishart distribution and the hyperboloid distribution are two special c ases in the class we define. We also give a unified view of several di stributions which are usually treated separately