Asymptotically minimax Bayes predictive densities

Abstract

Given a random sample from a distribution with density function that depends on an unknown parameter θ, we are interested in accurately estimating the true parametric density function at a future observation from the same distribution. The asymptotic risk of Bayes predictive density estimates with Kullback–Leibler loss function D(fθ||{f̂})=∫fθ log(fθ/f̂) is used to examine various ways of choosing prior distributions; the principal type of choice studied is minimax. We seek asymptotically least favorable predictive densities for which the corresponding asymptotic risk is minimax. A result resembling Stein’s paradox for estimating normal means by maximum likelihood holds for the uniform prior in the multivariate location family case: when the dimensionality of the model is at least three, the Jeffreys prior is minimax, though inadmissible. The Jeffreys prior is both admissible and minimax for one- and two-dimensional location problems.