Minimaxity in Predictive Density Estimation with Parametric Constraints

Abstract

This paper is concerned with estimation of a predictive density with parametric constraints under Kullback-Leibler loss. When an invariance structure is embedded in the problem, general and unified conditions for the minimaxity of the best equivariant predictive density estimator are derived. These conditions are applied to check minimaxity in various restricted parameter spaces in location and/or scale families. Further, it is shown that the generalized Bayes estimator against the uniform prior over the restricted space is minimax and dominates the best equivariant estimator in a location family when the parameter is restricted to an interval of the form [a0, ∞). Similar findings are obtained for scale parameter families. Finally, the presentation is accompanied by various observations and illustrations, such as normal, exponential location, and gamma model examples