密度估计的平滑惩罚反卷积（SPeD）
Smoothness-Penalized Deconvolution (SPeD) of a Density Estimate
AI总结：
本文提出一种平滑惩罚反卷积估计量，通过最小化带平滑惩罚的重建误差从受加性误差污染的观测中估计密度，建立了均方积分误差一致性及收敛速率，并证明其适用更广误差密度且优于核方法。
摘要：
本文解决从受已知密度的可加测量误差污染的观测值中估计平方可积概率密度的反卷积问题。该估计量从污染观测值的密度估计出发，最小化由积分平方$m$阶导数惩罚的重建误差。反卷积理论主要聚焦于基于核或小波的技术，但包括基于样条技术在内的其他方法及此平滑惩罚估计量在模拟研究中被发现优于核方法。本文通过为平滑惩罚方法建立渐近保证填补了其中一些空白。在平均积分平方误差中建立了一致性，并推导了Gaussian、Cauchy和Laplace误差密度的收敛速率，达到了文献中已有的一些下界。多数结果的假设较弱；该估计量可用于比反卷积核更广泛的误差密度类。我们的应用示例估计了随机抽样下某些细菌分离株的平均细胞毒性密度；该平均细胞毒性在实验中只能以可加误差测量，从而引发反卷积问题。我们还描述了一种用三次样条逼近解的方法，其可转化为一个二次规划。
英文摘要：
This paper addresses the deconvolution problem of estimating a square-integrable probability density from observations contaminated with additive measurement errors having a known density. The estimator begins with a density estimate of the contaminated observations and minimizes a reconstruction error penalized by an integrated squared $m$-th derivative. Theory for deconvolution has mainly focused on kernel- or wavelet-based techniques, but other methods including spline-based techniques and this smoothness-penalized estimator have been found to outperform kernel methods in simulation studies. This paper fills in some of these gaps by establishing asymptotic guarantees for the smoothness-penalized approach. Consistency is established in mean integrated squared error, and rates of convergence are derived for Gaussian, Cauchy, and Laplace error densities, attaining some lower bounds already in the literature. The assumptions are weak for most results; the estimator can be used with a broader class of error densities than the deconvoluting kernel. Our application example estimates the density of the mean cytotoxicity of certain bacterial isolates under random sampling; this mean cytotoxicity can only be measured experimentally with additive error, leading to the deconvolution problem. We also describe a method for approximating the solution by a cubic spline, which reduces to a quadratic program.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
