Posterior Distributions for Multivariate Normal Parameters

CORNISH (1961), following an earlier suggestion of Fisher (1954), has derived a joint fiducial distribution of the means of a p-variate normal distribution. The hyperellipsoidal fiducial region on the means given by this distribution is not the same as that given by the confidence region based on Hotelling's T2. In most earlier examples of differences between fiducial and confidence intervals, the disagreement has been obscured either by uncertainty as to the existence of a confidence interval based only on sufficient statistics, as in the Behrens-Fisher problem, or by the existence of two or more fiducial solutions, as in the ratio of two normal means. But the present disagreement is unequivocal since the confidence region is well defined and no other fiducial region has yet been produced to confound the issue. We have thought it of interest to examine this disagreement from the point of view of Bayes, using a prior distribution with an adjustable parameter, v. This examination leads to a posterior distribution of the means which reduces to the Fisher-Cornish density (F-C) for v = 2 and to what we will call the Hotelling density (H) for v = p + 1 (Section 2). We have investigated the effect of a choice of v on the marginal distribution of a single mean and on the marginal distribution of the elements of the covariance matrix (Section 3). The main result here is that there is no value of v that will lead to both Student's density as a marginal density for a single mean and Hotelling's density as the joint density for all the means. We also demonstrate that the posterior distribution of the elements of the covariance matrix in the bivariate case is different from the corresponding fiducial distribution for all values of v. Assuming we have N>p observations from a p-variate normal distribution with vector mean ,u' = (p1, ..., ,up) and covariance matrix E = {orij}, we then calculate from the observations the sample vector mean x' = (R, ..., p) and the sample covariance matrix S = {sj}, where