On optimal kernel choice for deconvolution

In this note we show that, from a conventional viewpoint, there are particularly close parallels between optimal-kernel-choice problems in non-parametric deconvolution, and their better-understood counterparts in density estimation and regression. However, other aspects of these problems are distinctly different, and this property leads us to conclude that “optimal” kernels do not give satisfactory performance when applied to deconvolution. This unexpected result stems from the fact that standard side conditions, which are used to ensure that the familiar kernel-choice problem has a unique solution, do not have statistically beneficial implications for deconvolution estimators. In consequence, certain “sub-optimal” kernels produce estimators that enjoy both greater efficiency and greater visual smoothness.