Some Common Structures of Distribution Problems in Geometric Probabilities and Multivariate Statistical Analysis

SYNOPTIC ABSTRACT

In a large number of situations one–to–one functions of the likelihood ratio test statistics connected with testing hypotheses on the parameters of one or more multivariate normal populations are distributed as the product of independent beta type–1 random variables when the hypotheses are assumed to be true. One–to–one functions of certain random volume contents generated by random points in a unit n–ball also have the structure of a product of independent beta type–1 random variables. Such common structures between multivariate statistical analysis and geometric probabilities are investigated in this article. After giving a survey of such problems, various techniques for handling the distributional aspects are discussed. A brief outline of the basic results is given. General techniques of getting asymptotic distributions are considered. A novel way of getting new results in mathematics by using statistical techniques, as a biproduct of the general procedures discussed, is also pointed out.

Key Words and Phrases:

• geometric probabilities
• distributions of random volumes
• product of beta variables
• general structures
• exact and asymptotic distributions