On the Exact Distribution of a Normalized Ratio of the Weighted Geometric Mean to the Unweighted Arithmetic Mean in Samples from Gamma Distributions

Abstract

The distribution of the product of several independent beta random variables as a mixture of beta distributions is derived by using a solution of Wilks's type B integral equation. This distribution is applied to finding the distribution of a normalized ratio of the weighted geometric mean (GM) to the unweighted arithmetic mean (AM) in random samples consisting of observations from gamma distributions, one observation being taken from each distribution. Two upper bounds for the truncation error related to the mixture representation are obtained. Application of these results to problems in testing statistical hypotheses is indicated.

Key Words:

• Gamma, beta, and Dirichlet distributions
• Wilks's type B integral equation
• Incomplete beta function
• Neyman-Pearson test criterion
• Bartlett's test statistic