Empirical Bayes deconvolution estimates

Abstract

An unknown prior density g(θ) has yielded realizations Θ1,⋯, ΘN. They are unobservable, but each Θi produces an observable value Xi according to a known probability mechanism, such as Xi ∼ Po(Θi). We wish to estimate g(θ) from the observed sample X1,⋯, XN. Traditional asymptotic calculations are discouraging, indicating very slow nonparametric rates of convergence. In this article we show that parametric exponential family modelling of g(θ) can give useful estimates in moderate-sized samples. We illustrate the approach with a variety of real and artificial examples. Covariate information can be incorporated into the deconvolution process, leading to a more detailed theory of generalized linear mixed models.