Asymptotically minimax Bayes predictive densities

• 成果类型: Article
• 署名作者: Aslan, Mihaela
• 署名单位: US Department of Veterans Affairs; Veterans Health Administration (VHA); VA Connecticut Healthcare System; Yale University
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/009053606000000885
• 发表日期: 2006
• 页码: 2921-2938
• 关键词: DISTRIBUTIONS

摘要

Given a random sample from a distribution with density function that depends on an unknown parameter 0, 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(f(theta)parallel to(f) over cap) = integral (f) over cap (theta) log (f(theta)/(f) over cap) 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.