Kernel methods and minimum contrast estimators for empirical deconvolution

TL;DR

This paper reviews kernel methods and minimum contrast estimators for nonparametric density deconvolution with measurement error, highlighting their connections and comparing their numerical performance.

Contribution

It reveals the equivalence of kernel and minimum contrast methods in density deconvolution and discusses the role of the sinc kernel in these approaches.

Findings

Kernel and minimum contrast methods yield identical results with fine grids.

Main differences are due to smoothing parameter selection.

Sinc kernel plays a central role in connecting the two approaches.

Abstract

We survey classical kernel methods for providing nonparametric solutions to problems involving measurement error. In particular we outline kernel-based methodology in this setting, and discuss its basic properties. Then we point to close connections that exist between kernel methods and much newer approaches based on minimum contrast techniques. The connections are through use of the sinc kernel for kernel-based inference. This `infinite order' kernel is not often used explicitly for kernel-based deconvolution, although it has received attention in more conventional problems where measurement error is not an issue. We show that in a comparison between kernel methods for density deconvolution, and their counterparts based on minimum contrast, the two approaches give identical results on a grid which becomes increasingly fine as the bandwidth decreases. In consequence, the main numerical differences between these two techniques are arguably the result of different approaches to choosing smoothing parameters.