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We consider nonparametric measurement error density deconvolution subject toheteroscedastic measurement errors as well as symmetry about zero and shapeconstraints, in particular unimodality. The problem is motivated byapplications where the observed data are estimated effect sizes fromregressions on multiple factors, where the target is the distribution of thetrue effect sizes. We exploit the fact that any symmetric and unimodal densitycan be expressed as a mixture of symmetric uniform densities, and model themixing density in a new way using a Dirichlet process location-mixture of Gammadistributions. We do the computations within a Bayesian context, describe asimple scalable implementation that is linear in the sample size, and show thatthe estimate of the unknown target density is consistent. Within ourapplication context of regression effect sizes, the target density is likely tohave a large probability near zero (the near null effects) coupled with aheavy-tailed distribution (the actual effects). Simulations show that unlikestandard deconvolution methods, our Constrained Bayesian Deconvolution methoddoes a much better job of reconstruction of the target density. Applications toa genome-wise association study (GWAS) and microarray data reveal similarresults.

Nonparametric Bayesian Deconvolution of a Symmetric Unimodal Density