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We develop a novel Empirical Bayes methodology for prediction under checkloss in high-dimensional Gaussian models. The check loss is a piecewise linearloss function having differential weights for measuring the amount ofunderestimation or overestimation. Prediction under it differs in fundamentalaspects from estimation or prediction under weighted-quadratic losses. Becauseof the nature of this loss, our inferential target is a pre-chosen quantile ofthe predictive distribution rather than the mean of the predictivedistribution. We develop a new method for constructing uniformly efficientasymptotic risk estimates which are then minimized to produce effective linearshrinkage predictive rules. In calculating the magnitude and direction ofshrinkage, our proposed predictive rules incorporate the asymmetric nature ofthe loss function and are shown to be asymptotically optimal. Using numericalexperiments we compare the performance of our method with traditional EmpiricalBayes procedures and obtain encouraging results.

Efficient Empirical Bayes prediction under check loss using Asymptotic Risk Estimates