On Predictive Density Estimation under α-Divergence Loss

Abstract

Based on X ∼ N_d(θ, σ_X²I_d), we study the efficiency of predictive densities under α-divergence loss L_α for estimating the density of Y ∼ N_d(θ, σ_Y²I_d). We identify a large number of cases where improvement on a plug-in density are obtainable by expanding the variance, thus extending earlier findings applicable to Kullback-Leibler loss. The results and proofs are unified with respect to the dimension d, the variances σ_X² and σ_Y², the choice of loss L_α; α ∈ (−1, 1). The findings also apply to a large number of plug-in densities, as well as for restricted parameter spaces with θ ∈ Θ ⊂ ℝ^d. The theoretical findings are accompanied by various observations, illustrations, and implications dealing for instance with robustness with respect to the model variances and simultaneous dominance with respect to the loss.