Deconvolution, convex optimization, non-parametric empirical Bayes and treatment of non-response

TL;DR

This paper introduces a computationally efficient deconvolution method for empirical Bayes estimation of unknown marginal distributions and functionals, with applications to non-response treatment, risk estimation, and false discovery rates.

Contribution

It develops a novel deconvolution approach using quadratic programming, improving scalability and applicability over traditional EM-based methods in empirical Bayes analysis.

Findings

Method performs well in simulations.

Applied successfully to Israeli labor survey data.

Provides confidence intervals for functionals of interest.

Abstract

Let $(Y_i,\theta_i)$, $i=1,...,n$, be independent random vectors distributed like $(Y,\theta) \sim G^*$, where the marginal distribution of $\theta$ is completely unknown, and the conditional distribution of $Y$ conditional on $\theta$ is known. It is desired to estimate the marginal distribution of $\theta$ under $G^*$, as well as functionals of the form $E_{G^*} h(Y,\theta)$ for a given $h$, based on the observed $Y_1,...,Y_n$. In this paper we suggest a deconvolution method for the above estimation problems and discuss some of its applications in Empirical Bayes analysis. The method involves a quadratic programming step, which is an elaboration on the formulation and technique in Efron(2013). It is computationally efficient and may handle large data sets, where the popular method, of deconvolution using EM-algorithm, is impractical. The main application that we study is treatment of non-response. Our approach is nonstandard and does not involve missing at random type of assumptions. The method is demonstrated in simulations, as well as in an analysis of a real data set from the Labor force survey in Israel. Other applications including estimation of the risk, and estimation of False Discovery Rates, are also discussed. We also present a method, that involves convex optimization, for constructing confidence intervals for $E_{G^*} h$, under the above setup.