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Two independent random samples of sizesN 1 andN 2 from multivariate normal populationsN p (θ1,∑1) andN p (θ2,∑2) are considered. Under the null hypothesisH 0: θ1=θ2, a single θ is generated from aN p(μ, Σ) prior distribution, while underH 1: θ1≠θ2 two means are generated from the exchangeable priorN p(μ,σ). In both cases Σ will be assumed to have a vague prior distribution. For a simple covariance structure, the Bayes factorB and minimum Bayes factor in favour of the null hypotheses is derived. The Bayes risk for each hypothesis is derived and a strategy is discussed for using the Bayes factor and Bayes risks to test the hypothesis.
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111 Test (1993) Vol. 2, No. 1-2, pp. 111-124 A Bayesian Approach to the Multivariate Behrens-Fisher Problem Under the Assumption of Proportional Covariance Matrices D. G. NEL and P. C. N. GROENEWALD Dept. Mathematical Statistics, University of the Orange Free State 339-Bloemfontein, 9300-South Africa SUMMARY Two independent random samples of sizes N1 and N~ from multivariate normal populations Np( 01, ~1) and Np( 02, ~2 ) are considered. Under the null hypothesis 11o : 01 = 02, a single 0 is generated from a Np(Iz, E) prior distribution, while under tll : 01 r 02 two means are generated from the exchangeable prior Nv(tz , or). In both cases E will be assumed to have a vague prior distribution. For a simple covariance structure, the Bayes factor 13 and minimum Bayes factor in favour of the null hypotheses is derived. The Bayes risk for each hypothesis is derived and a strategy is discussed for using the Bayes factor and Bayes risks to test the hypothesis. Keywords: BAYES RISK; HYPOTIIESIS TESTING; POSTERIOR PROBABIIJTY. 1. INTRODUCTION The well known Behrens-Fisher problem (Behrens (1929), Fisher (193