  We consider nonparametric measurement error density deconvolution subject to
heteroscedastic measurement errors as well as symmetry about zero and shape
constraints, in particular unimodality. The problem is motivated by
applications where the observed data are estimated effect sizes from
regressions on multiple factors, where the target is the distribution of the
true effect sizes. We exploit the fact that any symmetric and unimodal density
can be expressed as a mixture of symmetric uniform densities, and model the
mixing density in a new way using a Dirichlet process location-mixture of Gamma
distributions. We do the computations within a Bayesian context, describe a
simple scalable implementation that is linear in the sample size, and show that
the estimate of the unknown target density is consistent. Within our
application context of regression effect sizes, the target density is likely to
have a large probability near zero (the near null effects) coupled with a
heavy-tailed distribution (the actual effects). Simulations show that unlike
standard deconvolution methods, our Constrained Bayesian Deconvolution method
does a much better job of reconstruction of the target density. Applications to
a genome-wise association study (GWAS) and microarray data reveal similar
results.
