Properties of the singular, inverse and generalized inverse partitioned Wishart distributions

In this paper we discuss the distributions and independency properties of several generalizations of the Wishart distribution. First, an analog to Muirhead [R.J. Muirhead, Aspects of Multivariate Statistical Theory, Wiley, New York, 1982] Theorem 3.2.10 for the partitioned matrix A=(Aij)i,j=1,2 is established in the case of arbitrary partitioning for singular and inverse Wishart distributions. Second, the density of A21A11−1 is derived in the case of singular, non-central singular, inverse and generalized inverse Wishart distributions. The importance of the derived results is illustrated with an example from portfolio theory.