Nonparametric Empirical Bayes Tweedie's Estimator for Normal Means with Heteroscedastic Errors

• Activity Number: 584 - Advances in Semi- and Nonparametric Statistical Analysis
• Type: Contributed
• Date/Time: Wednesday, August 1, 2018 : 2:00 PM to 3:50 PM
• Sponsor: IMS
• Author(s): Luella Fu* and Gareth James and Wenguang Sun
• Companies: USC Marshall School of Business and USC Marshall School of Business and USC Marshall School of Business
• Keywords: compound decision; empirical Bayes; kernel smoothing; shrinkage estimation; SURE; Tweedie's formula

Abstract

The problem of estimating a vector of normal means, subject to a fixed variance term, has been extensively studied, but many practical situations instead involve a heterogeneous variance. Hence, we consider the problem of estimating a vector of normal means with heteroscedastic variances and propose the "Nonparametric Empirical Bayes SURE Tweedie's" (NEST) estimator. NEST estimates the marginal density of the data, for any pair of x and its variance, using a smoothing kernel that weights observations according to their distance from both x and its variance. NEST then applies the estimated density to a generalized version of Tweedie's formula to estimate the corresponding mean vector. NEST is simple to calculate but flexible enough to accommodate general settings. Additionally, a Stein-type unbiased risk estimate (SURE) criterion is developed to select NEST's tuning parameters. Our theoretical results show that NEST is asymptotically optimal, while simulation studies show that it outperforms competitive methods, with substantial efficiency gains in many settings. The method is further demonstrated on a data set measuring school performance gaps.