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

TL;DR

This paper demonstrates that consistent estimation of the mean of a mixing distribution in a non-parametric empirical Bayes framework is possible without the usual identifiability assumptions, with applications to non-response treatment.

Contribution

It introduces a method for consistent estimation of the mixing distribution mean without requiring identifiability, addressing challenges in non-response scenarios.

Findings

Consistency demonstrated in simulations

Estimation feasible without identifiability assumptions

Applicable to non-response and missing data situations

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.