On transformations and determinants of Wishart variables on symmetric cones

Wishart distributions on the cone of real symmetric positive-definite matrices are a well-studied subject in statistics. Recently, these distributions have been defined on the symmetric cone of a simple Euclidean Jordan algebra (formerly called ``the domain of positivity of a simple formally-real Jordan algebra''). In this paper we extend some known properties of the Wishart distribution on real symmetric matrices to the setting of simple Euclidean Jordan algebras. These are a formula for the expected value of the determinant (= reduced norm) of a sum of two Wishart variables and the determination of the distribution of the Peirce components of a Wishart variable. Our statistical results are based on some new purely algebraic results on the determinant of Euclidean Jordan algebras.

This paper has appeared in J. Theoret. Probab. 10 (1997), 867--902.

H. Massam <massamh@mathstat.yorku.ca>

E. Neher