A simulation approach to nonparametric empirical Bayes analysis

We deal with general mixture or hierarchical models of the form m(x) = inte gral (Theta) f(x \ theta )g(theta )d theta, where g(B) and m(x) are called mixing and mixed or compound densities respectively, and B is called the mi xing parameter The usual statistical application of these models emerges wh en we have data x(i), i = i,..., n with densities f(x(i) \ theta (i)) for g iven theta (i), and the theta (i) are independent with common density g(B), For a certain well known class of densities f(x \ theta), we present a sam ple-based approach to reconstruct g(B), We first provide theoretical result s and then we use, in an empirical Bayes spirit, the first four moments of the data to estimate the first four moments of g(B). By using sampling tech niques we proceed in a fully Bayesian fashion to obtain any posterior summa ries of interest. Simulations which investigate the operating characteristi cs of our proposed methodology are presented. We illustrate our approach us ing data from mixed Poisson and mixed exponential densities.