Goodness of Prediction Fit for Multivariate Linear Models

Using both frequentist and Bayesian techniques, predicting densities are derived for future observations from a multivariate linear model with matrix normal error terms. All the candidates belong to a general location-scale family of predicting densities. An analytic comparison is undertaken, using a Kullback—Leibler loss, by citing an optimal member of a subclass including most of these predicting densities as members. The subclass is based on an invariant Student-t random matrix, and the optimal member is the Bayesian predictive density corresponding to a Jeffreys noninformative prior. Information-based numerical comparisons illustrate the nature of the dominance.