On construction of consistent mixing distributions for compound decisions

Deely and Kruse [Deely, J.J., Kruse, R.L., 1968. Construction of sequences estimating the mixing distribution. The Annals of Mathematical Statistics 39 (1), 286-288] proposed a minimum distance method of estimating the prior distribution of an empirical Bayes decision problem and showed their constructed distributions converge weakly to the prior distribution, on a set of probability 1, when the component distribution function is continuous. The advantage of their method is that the proposed estimates can be easily computed by linear programming software. In this paper we extend the range of applications of their method to compound decision theory by establishing a desirable L1-consistency of the constructed estimators. The extension that we obtain does not require continuity of the component distribution functions.