A Simulation Approach to Nonparametric Empirical Bayes Analysis
Πέτρος Δελλαπόρτας, Dimitris Karlis · International Statistical Review · 2001
Summary We deal with general mixture of hierarchical models of the formm(x) = føf(x |θ) g (θ)dθ, whereg(θ)andm(x)are called mixing and mixed or compound densities respectively, and θ is called the mixing parameter. The usual statistical application of these models emerges when we have dataxi, i = 1,…,nwith densitiesf(xi|θi)for given θi, and the θ1are independent with common densityg(θ). For a certain well known class of densitiesf(x |θ), we present a sample‐based approach to reconstructg(θ). We first provide theoretical results and then we use, in an empirical Bayes spirit, the first four moments of the data to estimate the first four moments ofg(θ). By using sampling techniques we proceed in a fully Bayesian fashion to obtain any posterior summaries of interest. Simulations which investigate the operating characteristics of our proposed methodology are presented. We illustrate our approach using data from mixed Poisson and mixed exponential densities.