A Bayesian solution to the multivariate Behrens-Fisher problem

Abstract

One of the most common problems in applied statistics is to compare two normal population means when the ratio of variances is unknown and not equal to 1. This is the known Behrens–Fisher problem. There are many approaches to the distribution to the t-statistic in the univariate circumstance under the Behrens–Fisher problem. In the multivariate case, most solutions are based on adjusting the degrees of freedom to obtain better approximations to the chi-squared or Hotelling’s T2 distributions. In both circumstances there are Bayesian solutions proposed by some authors. This work aimed to propose a computational Bayesian solution to the multivariate Behrens–Fisher problem based on the complex analytical solution of Johnson and Weerahandi (1988), to evaluate its performance through Monte Carlo simulation computing the type I error rates and power and to compare it with the modified Nel and Van der Merwe test, that is considered the best frequentist solution. The inferences were made to the population difference δ of the mean vectors. It was used as a conjugate prior distribution to the population mean vector (μi) and covariance matrix (Σi) obtaining a posterior multivariate t distribution to μi, for i=1, 2. In general, the Bayesian test was conservative for samples of different sizes and liberal in some circumstances of equal and small sample sizes and its power was equal to or greater than that of its competitor in large samples and/or in balanced circumstances. The new solution has competitive advantages and in some circumstances surpasses its main competitor, therefore its use in real cases should be recommended.