Generalized maximum likelihood estimation of the mean of parameters of mixtures. With applications to sampling and to observational studies

Abstract

Let $f(y\mid \mathit{\theta }),\mathit{\theta }\in \mathrm{\Omega }$ be a parametric family, $\mathrm{\eta }(\mathit{\theta })$ a given function, and G an unknown mixing distribution. It is desired to estimate $E_G(\mathrm{\eta }(\mathit{\theta }))\equiv {\mathrm{\eta }}_G$ based on independent observations $Y_1,\mathrm{...},Y_n$, where $Y_i\sim f(y\mid {\mathit{\theta }}_i)$, and ${\mathit{\theta }}_i\sim G$ 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 [14], with implications to a corresponding setup in which ${\mathit{\theta }}_1,\mathrm{...},{\mathit{\theta }}_n$ are fixed parameters. We also relate the above problem, of estimating ${\mathrm{\eta }}_G$, to nonparametric 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.