Density deconvolution is an ill-posed inverse problem because recovering the latent density amplifies high-frequency variation in the observed data. We propose a two-stage likelihood procedure. The first estimator maximizes the convolution likelihood over a regularized Gaussian-mixture sieve; the second treats this fit as an empirical prior and averages the fitted conditional latent densities. Under fixed-model misspecification, the estimators generally have different population targets. For bounded local multiplicative perturbations of the latent density, the population posterior map is locally contractive in chi-square divergence under identifiable convolution. Under correct finite-dimensional specification, however, reconstruction adds a nonnegative first-order variance component. For growing sieves, we separate observed-domain estimation, inverse stability, and posterior empirical variation. We derive explicit rates under Gaussian error and recover the classical ordinary-smooth exponent, up to logarithmic factors, with a constructive verification for Laplace error. Simulations and a Framingham blood-pressure application provide empirical performance evidences.
M. Di Marzio, S. Fensore, Chiara Passamonti et al.· 0 citations
Density deconvolution is an ill-posed inverse problem, as recovering the latent distribution amplifies high-frequency noise in the observed data. We propose a two-stage likelihood-based procedure. The first estimator maximizes the convolution likelihood over a regularized Gaussian-mixture sieve, while the second reuses the fitted density as an empirical prior and averages the resulting conditional latent densities over the observations. The two estimators have different population targets under misspecification: posterior reconstruction locally contracts misspecification bias under identifiable convolution, but introduces an additional first-order variance component under correct finite-dimensional specification. We establish observed- and latent-domain convergence rates under Gaussian error. Under ordinary-smooth error, the direct estimator in $L^2$ and the posterior reconstruction in $L^1$ attain the classical deconvolution exponent, up to logarithmic factors. Numerical experiments and an application to Framingham blood-pressure data illustrate their complementary finite-sample behavior.
M. Marzio, S. Fensore, Chiara Passamonti et al.· 0 citations
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