Empirical Bayes prediction for the multivariate newsvendor loss function

Motivated by an application in inventory management, we consider the multi-product newsvendor problem of finding the optimal stocking levels that minimize the total backorder and lost sales costs. We focus on a setting where we have a large number of products and observe only noisy estimates of the underlying demand. We develop an Empirical Bayes methodology for predicting stocking levels, using data-adaptive linear shrinkage strategies which are constructed by minimizing uniformly efficient asymptotic risk estimates. In calculating the magnitude and direction of shrinkage, our proposed predictive rules incorporate the asymmetric nature of the piecewise linear newsvendor loss function and are shown to be asymptotically optimal. Using simulated data, we study the non-asymptotic performance of our method and obtain encouraging results. The asymptotic risk estimation method and the proof techniques developed here can also be used to formulate optimal empirical Bayes predictive strategies for general piecewise linear and related asymmetric loss functions.