Type-1 Beta Distribution and its Connections to Likelihood Ratio Test - Nicy Sebastian; T. Princy (Statistica)
In many cases involving hypothesis testing for parameters in multivariate Gaussian populations and certain other populations, likelihood ratio criteria, or their one-to-one functions, can be expressed in terms of the determinant of a real type-1 beta matrix. In geometrical probability problems, when the random points are type-1 beta distributed, the volume content of the parallellotope generated by these points is also associated with the determinant of a real type-1 beta matrix. These problems in the corresponding complex domain do not seem to have been discussed in the literature. It is well-known that the determinant of a real type-1 beta matrix can be written as a product of statistically independently distributed real scalar type-1 beta random variables. This paper addresses the general h-th moments of a scalar random variable, in either the real or complex domain, for any arbitrary h. The structure of these moments is quite general, and the paper provides exact distribution results, asymptotic gamma function results, and asymptotic normal results for both the real and complex domains.