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

We consider a Nonparametric Empirical Bayes (NPEB) framework. Let $Y_i$ be random variables, $Y_i \sim f(y|θ_i)$, $i=1,...,n$, where $θ_i \sim G$, and $θ_i \in Θ$ are independent. The variables $Y_i $ are conditionally independent given $θ_i, \; i=1,...,n$. The mixing distribution $G$ is unknown and assumed to belong to a nonparametric class $\{G \}$. Let $η(θ)$ be a function of $θ$. We address the problem of consistently estimating $E_G η(θ) \equiv η_G$. 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.