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.