Empirical Bayes deconvolution estimates

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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.

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