Statistica Sinica 23 (2013), 333-357

doi:http://dx.doi.org/10.5705/ss.2011.071

EMPIRICAL BAYES IN THE PRESENCE

OF EXPLANATORY VARIABLES

Noam Cohen$^1$, Eitan Greenshtein$^1$ and Ya'acov Ritov$^2$

$^1$Israeli CBS and $^2$The Hebrew University

Abstract: 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).

Key words and phrases: Compound decision, empirical Bayes.