Empirical bayes estimation of functionals of unknown probability measures

The empirical Dayes approach to one and two sal-npie problcrns has beeir considered by Korwar and Hollander (1976), Holiander and Korwar (1976) and Phadia and Susarla (1979). In this article we essen- tially generalize their empirical Bayes results by replacing the inlicaro-functions of. the sets (?∞,x) and {X≦Y} by arbitrary mea5, irable functions h(x) and h(x,y). More speclfically, the ernpiricaion yes estimation of esrimabie paramerers of degree one ani KG,I;ti kliown probability measure Pon (R,R) is considered. The asymptotic optimality of the these estimators, obtaining the exact risk expressions, is established. Also the results of Dalal and Phad (1983) we extended to the estimation of an estimable parametric function of an unknow probability measure P on (R² , B²)