Observe n independent normal samples, each with their own mean value. For each sample a new independent observation is taken, having the same distribution as the observed sample values. The objective is to predict this new value under a check-loss error measure. [“Check-loss” is linear in either under- or over- estimation, but with different multiplicative constant for each of the two possible types of error.] This is equivalent to a desire to estimate a pre-specified quantile of the predictive distribution. In terms of applications this setting is equivalent to the traditional newsvendor problem, with interest focused on a big data regime incorporating many (n) different products. We develop and use an Empirical Bayes methodology that minimizes a new, uniformly efficient asymptotic risk estimate.

In common with many other problems we find that empirical Bayes shrinkage provides better performance than simple coordinate-wise rules. However, the problem here differs in fundamental respects from estimation or prediction under the weighted quadratic losses considered in most of the previous literature. This necessitates different strategies for creation of effective empirical Bayes predictors. The hyper-parameter estimator we develop involves an appropriate use of Hermite polynomial expansions for the relevant stochastic functions.

Introduction:

Dr.Lawrence D. Brown is a Miers Busch Professor and Professor of Statistics at The Wharton School, University of Pennsylvania. His research Areas are Statistical decision theory, Statistical inference, Nonparametric function estimation, Foundations of statistics, Sampling theory (census data), Empirical queueing science.

//www-stat.wharton.upenn.edu/~lbrown/

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