Consistent Empirical Bayes estimation of the mean of a mixing distribution without identifiability assumption. With applications to treatment of non-response

Abstract

{\bf Abstract} Consider a Non-Parametric Empirical Bayes (NPEB) setup. We observe $Y_i,\sim f(y|{\theta }_i)$, ${\theta }_i\in \Theta$ independent, where ${\theta }_i\sim G$ are independent $i=1,...,n$. The mixing distribution $G$ is unknown $G\in \{G\}$ with no parametric assumptions about the class $\{G\}$. The common NPEB task is to estimate ${\theta }_i,i=1,...,n$. Conditions that imply 'optimality' of such NPEB estimators typically require identifiability of $G$ based on $Y_1,...,Y_n$. We consider the task of estimating $E_G\theta$. We show that `often' consistent estimation of $E_G\theta$ is implied without identifiability. We motivate the later task, especially in setups with non-response and missing data. We demonstrate consistency in simulations.

17 pages