The use of the disguised Wishart distribution in a Bayesian approach to tolerance region construction

Summary In attacking the problem of this paper (see Section 1), the authors were confronted with finding the distribution of a (k×k) matrix of random variablesR=P′VP, wherePP′=Σ ^-1, and where the (k×k) symmetric matrix Σ^-1 has the Wishart distribution, matrix [(n−1)V]⁻¹, and degrees of freedom (n−1), withV a (k×k) symmetric positive definite matrix of constants. This distribution (whenP is lower triangular with positive diagonal elements), and a related result, has recently been found by the authors and given in Tan and Guttman [7]. In this paper we use these results (stated here without proof in Theorems 1.1 and 1.2) to help us construct a β-expectation tolerance region, when sampling is from thek-variate normal,N(μ,Σ), where Σ is positive definite. This research was partially supported by the National Institute of Health under Grant No. GM 15422, and the Wisconsin Research Foundation.