Compound decision in the presence of proxies

We study the problem of incorporating covariates in a compound decision setup. It is desired to estimate the means of n response variables, which are indepen- dent and normally distributed, and each is 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 numeri- cally with Fay-Herriot and other methods, in a real data situation, where the goal is to estimate certain proportions in many small areas (Statistical-Areas). We also demonstrate our approach through the baseball data set, originally analayzed by Brown(2010).