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.

中文翻译：

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α-散度损失下的预测密度估计

基于X〜n的_d（θ，σ ² _X我_d）中，我们研究预测密度下的效率α-发散损耗大号_α用于估计的密度Y〜Ñ _d（θ，σ ² _ÿ我_d）。我们确定了许多情况，可以通过扩大方差来提高插件密度，从而扩展了适用于Kullback-Leibler损失的早期发现。结果和证明是统一的相对于所述尺寸d，方差σ ² _X和σ ² _ÿ ，损失的选择大号_α ; α＆Element;（ - 1，1）。这些发现也适用于大量的插件密度，以及用于与受限参数空间θ＆Element;Θ⊂ℝ ^d。理论发现伴随着各种观察，说明和影响，例如关于模型方差的稳健性和关于损失的同时优势。