ABSTRACT
This chapter discusses several generalizations which lead to matrix variate analogs of beta type I and type II distributions. It defines the matrix variate beta distributions of type I and type II. The matrix variate beta type II distribution is also known as matrix variate F-distribution. These distributions belong to the class of orthogonally invariant and residual independent distributions. The chapter provides derivations of beta distributions of type I and II, generalizing the ratios xy and xx+y to the matrix case. It presents some properties of the random matrices distributed as matrix variate beta type I and II. The chapter gives various factorizations of beta type I and type II matrices. Like Wishart matrix which factorizes into normal matrices, the beta type I and type II matrices factorize into inverted t- and t- matrices. The chapter also provides some distributions related to the matrix variate beta type I and type II distributions.
