Exact minimax strategies for predictive density estimation, data compression, and model selection

For location and scale families of distributions and related settings of linear regression, we determine minimax procedures for predictive density estimation, for universal data compression, and for the minimum description length (MDL) criterion for model selection. The analysis gives the best invariant and indeed minimax procedure for predictive density estimation by directly verifying extended Bayes properties or, alternatively, by general aspects of decision theory on groups which are shown to simplify in the case of Kullback-Leibler loss. An exact minimax rule is generalized Bayes using a uniform (Lebesgue measure) prior on the location and log-scale parameters, which is made proper by conditioning on an initial set of observations.