Smoothness-Penalized Deconvolution: Rates of Convergence, Choice of Tuning Parameter, and Inference
This work 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 work fills in some of the gaps in theory by proving rates of convergence for the smoothness-penalized estimator and its spline approximant. We contribute to the practical use of the estimator by developing an unbiased estimator of its risk, along with a method for using that estimated risk to choose the tuning parameter; this outperforms the SURE-based tuning parameter method that has been proposed for a similar estimator. Finally, we develop methods for constructing bias-corrected pointwise confidence intervals and assess the coverage properties in a simulation study, finding that they have uniformly lower coverage error than the naive normal-theory intervals.