Density Deconvolution with Additive Measurement Errors using Quadratic Programming (revision)

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 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.

噪声数据密度反卷积的分布估计是一项众所周知的技术难题，尤其是在诸如高斯分布这类典型的噪声分布中。本研究开发了一种基于二次规划（QP）的密度反卷积估计器，其估计效果优于核密度反卷积方法。QP方法在正则化权衡方面显示出更为有利的特性，即在分布的尾部，相较于过度平滑与振荡之间。此外，它能够简便地整合多种常见的密度约束条件，如非负性、积分归一化、单峰性、尾部凸性、尾部单调性和支持约束。我们证实，与现有方法相比，QP方法在估计性能方面具有优势。当仅引入普遍适用的非负性和积分归一化约束时，其性能尤为出色；在适用的情况下，进一步引入其他常见约束（例如非负支持、单峰性、尾部单调性或凸性等）可显著提升估计效果。补充材料可在网上获取，包括R代码、QPdecon R包、QPdecon包的vignette、用作示例的sodium数据集以及包含证明和额外图表的附录。