Generalized maximum likelihood estimation of the mean of parameters of mixtures, with applications to sampling

Let $f(y|θ), \; θ\in Ω$ be a parametric family, $η(θ)$ a given function, and $G$ an unknown mixing distribution. It is desired to estimate $E_G (η(θ))\equiv η_G$ based on independent observations $Y_1,...,Y_n$, where $Y_i \sim f(y|θ_i)$, and $θ_i \sim G$ are iid. We explore the Generalized Maximum Likelihood Estimators (GMLE) for this problem. Some basic properties and representations of those estimators are shown. In particular we suggest a new perspective, of the weak convergence result by Kiefer and Wolfowitz (1956), with implications to a corresponding setup in which $θ_1,...,θ_n$ are {\it fixed} parameters. We also relate the above problem, of estimating $η_G$, to non-parametric empirical Bayes estimation under a squared loss. Applications of GMLE to sampling problems are presented. The performance of the GMLE is demonstrated both in simulations and through a real data example.

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