Asymptotically minimax Bayes predictive densities

Given a random sample from a distribution with density function that depends on an unknown parameter &thetas; = (&thetas; ₁,…, &thetas;_p), we are interested in accurately estimating the true parametric density function at a future observation from the same distribution.; Asymptotic expressions for the marginal and Bayes predictive densities and the Kullback-Leibler divergence are obtained at the true value of the parameter. These expressions involve log-likelihood and log-prior derivatives and the expectations of their products, all evaluated at the true value.; The asymptotic risk of Bayes predictive density estimates with Kullback-Leibler loss function D(f&thetas;|| fˆ) = ∫ log (f&thetas;/ fˆ)f&thetas; is used to examine various ways of choosing prior distributions; the principal type of choice studied is minimax. We will seek asymptotically least favorable predictive densities for which the corresponding asymptotic risk is minimax.; The asymptotic risk of Bayes estimators has a term depending on the prior which is identical with the prior term in the asymptotic risk expression of Bayes predictive density estimates. In cases where a transitive invariant group applies, it may be shown that the asymptotic risk of the Bayes estimate and the predictive density risk differ by a constant; so minimaxity of one implies minimaxity of the other.; A result resembling Stein's paradox for estimating normal means by the maximum likelihood holds for the uniform prior in the multivariate location family case: when the dimensionality of the model is at least three, Jeffreys' prior is minimax, though inadmissible. For the one- and two-dimensional case the Jeffreys prior is both admissible and minimax. These admissibility and minimaxity results were determined using differential geometry and partial differential equations, in particular potential theory.; For the multivariate normal scale case we prove that within the α-class of relatively invariant priors (Hartigan, 1964) Jeffreys' prior produces the smallest constant risk, and so is best among the α-class of priors.; In the univariate normal scale case Jeffreys' prior is also minimax; appropriately reparametrizing the problem we fall onto the univariate location case where we already have proven the uniform prior is minimax.; For the univariate normal location-scale case the optimal relatively invariant prior within the α-class of priors is not the Jeffreys prior.