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Let $f(y|\theta), \; \theta \in \Omega$ be a parametric family,$\eta(\theta)$ a given function, and $G$ an unknown mixing distribution. It isdesired to estimate $E_G (\eta(\theta))\equiv \eta_G$ based on independentobservations $Y_1,...,Y_n$, where $Y_i \sim f(y|\theta_i)$, and $\theta_i \simG$ are iid.  We explore the Generalized Maximum Likelihood Estimators (GMLE) for thisproblem. Some basic properties and representations of those estimators areshown. In particular we suggest a new perspective, of the weak convergenceresult by Kiefer and Wolfowitz (1956), with implications to a correspondingsetup in which $\theta_1,...,\theta_n$ are {\it fixed} parameters. We alsorelate the above problem, of estimating $\eta_G$, to non-parametric empiricalBayes estimation under a squared loss.  Applications of GMLE to sampling problems are presented. The performance ofthe GMLE is demonstrated both in simulations and through a real data example.

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