Efficient Empirical Bayes Prediction Under Check Loss Using Asymptotic Risk Estimates

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