On Predictive Density Estimation under α-divergence Loss
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Based on X ∼ N_d(θ, σ^2_X I_d), we study the efficiency of predictive densities under α-divergence loss L_α for estimating the density of Y ∼ N_d(θ, σ^2_Y I_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^2_X and σ^2_Y, 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.
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