Density Deconvolution With Additive Measurement Errors Using Quadratic Programming

Abstract

Distribution estimation for noisy data via density deconvolution is a notoriously difficult problem, especially for typical noise distributions like Gaussian. We develop a density deconvolution estimator based on quadratic programming (QP) that can achieve better estimation than kernel density deconvolution methods. The QP approach appears to have a more favorable regularization tradeoff between oversmoothing versus oscillation, especially at the tails of the distribution. An additional advantage is that it is straightforward to incorporate a number of common density constraints such as nonnegativity, integration-to-one, unimodality, tail convexity, tail monotonicity, and support constraints. We demonstrate that the QP approach has favorable estimation performance relative to existing methods. Its performance is superior when only the universally applicable nonnegativity and integration-to-one constraints are incorporated, and incorporating additional common constraints when applicable (e.g., nonnegative support, unimodality, tail monotonicity or convexity, etc.) can further substantially improve the estimation. Supplementary materials for this article are available online and include R code, the R package QPdecon, a vignette for the QPdecon package, the sodium dataset that is used as an example, and appendices with a proof and additional figures.

Keywords:

• Additive error model
• Gaussian and Laplace distributed noise
• Nonparametric density estimation
• Shape constraints

Acknowledgments

We thank Professor Abhra Sarkar for giving us the BayesME package and Professor Aurore Delaigle for informing us that the deconvolute package was on GitHub.

Additional information

Funding

This work was supported in part by NSF grant CMMI-1436574, which the authors gratefully acknowledge.