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 prameters.

Keywords bandwidth, inverse problems, kernel estimators, local linear methods, local polynomial methods, minimum contrast methods, non-parametric curve estimation, nonparametric density estimation, non-parametric regression, penalised contrast methods, rate of convergence, sinc kernel, statistical smoothing

AMS subject classification (MSC2010) 62G08, 62G051

Introduction

Summary

Our aim in this paper is to give a brief survey of kernel methods for solving problems involving measurement error, for example problems involving density deconvolution or regression with errors in variables, and to relate these ‘classical’ methods (they are now about twenty years old) to new approaches based on minimum contrast methods.