Empirical bayes in the presence of explanatory variables

We study the problem of incorporating covariates in a compound decision setup. It is desired to estimate the means of n response variables that are independent and normally distributed, each accompanied by a vector of covariates. We suggest a method that involves non-parametric empirical Bayes techniques and may be viewed as a generalization of the celebrated Fay-Herriot (1979) method. Some optimality properties of our method are proved. We also compare it numerically with Fay-Herriot and other methods, in a real data situation where the goal is to estimate certain proportions in many small areas. We also demonstrate our approach through the baseball data set originally analyzed by Brown (2008).