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

Let $(Y_i,θ_i)$, $i=1,...,n$, be independent random vectors distributed like $(Y,θ) \sim G^*$, where the marginal distribution of $θ$ is completely unknown, and the conditional distribution of $Y$ conditional on $θ$ is known. It is desired to estimate the marginal distribution of $θ$ under $G^*$, as well as functionals of the form $E_{G^*} h(Y,θ)$ 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.

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