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.

翻译：本文研究反卷积问题，即从含有已知密度加性测量误差的观测中估计平方可积概率密度。该估计量始于对受污染观测的密度估计，并通过最小化受积分平方m阶导数惩罚的重构误差进行估计。反卷积理论主要集中于基于核或小波的技术，但包括样条方法及本文提出的光滑性惩罚估计量在内的其他方法，已在模拟研究中展现出优于核方法的性能。本文通过建立光滑性惩罚方法的渐近保证，填补了部分理论空白。我们证明了均方积分误差的一致性，并导出了高斯、柯西和拉普拉斯误差密度的收敛速率，这些速率达到了文献中已有的部分下界。多数结果的假设条件较弱；该估计量可适用于比反卷积核更广泛的误差密度族。在应用实例中，我们估计了随机抽样下某些细菌分离株平均细胞毒性的密度；该平均细胞毒性只能通过加性误差的实验测量获得，从而构成反卷积问题。我们还描述了一种通过三次样条近似求解的方法，该方法可简化为二次规划问题。