Generalized Maximum Likelihood Estimators and their applications to stratified sampling and post-stratification with many unobserved strata
Consider the problem of estimating a weighted average of the means of $n$ strata, based on a random sample with realized $K_i$ observations from stratum $i, \; i=1,...,n$. This task is non-trivial in cases where for a significant portion of the strata the corresponding $K_i=0$. Such a situation may happen in post-stratification, when it is desired to have a very fine sftratification. A fine stratification could be desired in order that assumptions, or, approximations, like Missing At Random conditional on strata, will be appealing. A fine stratification could also be desired in observational studies, when it is desired to estimate average treatment effect, by averaging the effects in small and homogenous strata. Our approach is based on applying Generalized Maximum Likelihood Estimators (GMLE), and ideas that are related to Non-Parametric Empirical Bayes, in order to estimate the means of strata $i$ with corresponding $K_i=0$. There are no assumptions about a relation between the means of the unobserved strata (i.e., with $K_i=0$) and those of the observed strata. The performance of our approach is demonstrated both in simulations and on a real data set. Some consistency and asymptotic results are also presented. In addition, related basic results about GMLE estimation of the mean of mixtures of exponential families are provided.