Nonparametric Empirical Bayes Estimation on Heterogeneous Data.

The simultaneous estimation of many parameters $\eta_i$, based on a corresponding set of observations $x_i$, for $i=1,\ldots, n$, is a key research problem that has received renewed attention in the high-dimensional setting. %The classic example involves estimating a vector of normal means $\mu_i$ subject to a fixed variance term $\sigma^2$. However, Many practical situations involve heterogeneous data $(x_i, \theta_i)$ where $\theta_i$ is a known nuisance parameter. Effectively pooling information across samples while correctly accounting for heterogeneity presents a significant challenge in large-scale estimation problems. We address this issue by introducing the "Nonparametric Empirical Bayes Smoothing Tweedie" (NEST) estimator, which efficiently estimates $\eta_i$ and properly adjusts for heterogeneity %by approximating the marginal density of the data $f_{\theta_i}(x_i)$ and applying this density to via a generalized version of Tweedie's formula. NEST is capable of handling a wider range of settings than previously proposed heterogeneous approaches as it does not make any parametric assumptions on the prior distribution of $\eta_i$. The estimation framework is simple but general enough to accommodate any member of the exponential family of distributions. %; a thorough study of the normal means problem subject to heterogeneous variances is presented to illustrate the proposed framework. Our theoretical results show that NEST is asymptotically optimal, while simulation studies show that it outperforms competing methods, with substantial efficiency gains in many settings. The method is demonstrated on a data set measuring the performance gap in math scores between socioeconomically advantaged and disadvantaged students in K-12 schools.