  Let $f(y|\theta), \; \theta \in \Omega$ be a parametric family,
$\eta(\theta)$ a given function, and $G$ an unknown mixing distribution. It is
desired to estimate $E_G (\eta(\theta))\equiv \eta_G$ based on independent
observations $Y_1,...,Y_n$, where $Y_i \sim f(y|\theta_i)$, and $\theta_i \sim
G$ 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 Kiefer and Wolfowitz (1956), with implications to a corresponding
setup in which $\theta_1,...,\theta_n$ are {\it fixed} parameters. We also
relate the above problem, of estimating $\eta_G$, 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.
