Consistent Empirical Bayes Estimation of the Mean of a Mixing Distribution with Applications to Treatment of Nonresponse

Abstract

We consider a Nonparametric Empirical Bayes (NPEB) framework. Let $Y_i$ be random variables, $Y_i\sim f(y|{\theta }_i)$, $i=1,...,n$, where ${\theta }_i\sim G$, and ${\theta }_i\in \Theta$ are independent. The variables $Y_i$ are conditionally independent given ${\theta }_i,i=1,...,n$. The mixing distribution $G$ is unknown and assumed to belong to a nonparametric class $\{G\}$. Let $\eta (\theta )$ be a function of $\theta$. We address the problem of consistently estimating $E_G\eta (\theta )\equiv {\eta }_G$. This problem becomes particularly challenging when $G$ cannot be consistently estimated from the observed data. We motivate this problem, especially in contexts involving nonresponse and missing data. For such cases, a consistent estimation method is suggested and its performance is demonstrated through simulations.