Large-Scale Estimation under Unknown Heteroskedasticity

This paper studies nonparametric empirical Bayes methods in a heterogeneous parameters framework that features unknown means and variances. We provide extended Tweedie's formulae that express the (infeasible) optimal estimators of heterogeneous parameters, such as unit-specific means or quantiles, in terms of the density of certain sufficient statistics. These are used to propose feasible versions with nearly parametric regret bounds of the order of $(\log n)^\kappa / n$. The results rely on a distributional assumption, and thus a misspecification analysis is also presented. The estimators are employed in a study of teachers' value-added, where we find that allowing for heterogeneous variances across teachers is crucial for delivery optimal estimates of teacher quality and detecting low-performing teachers.