A Design-Adaptive Local Polynomial Estimator for the Errors-in-Variables Problem

Abstract

Local polynomial estimators are popular techniques for nonparametric regression estimation and have received great attention in the literature. Their simplest version, the local constant estimator, can be easily extended to the errors-in-variables context by exploiting its similarity with the deconvolution kernel density estimator. The generalization of the higher order versions of the estimator, however, is not straightforward and has remained an open problem for the last 15 years. We propose an innovative local polynomial estimator of any order in the errors-in-variables context, derive its design-adaptive asymptotic properties and study its finite sample performance on simulated examples. We provide not only a solution to a long-standing open problem, but also provide methodological contributions to error-invariable regression, including local polynomial estimation of derivative functions.

Predictive Inference for Integrated Volatility

“…Recently, Delaigle, Fan and Carroll (2009) have derived asymptotic normality results for local polynomial estimators. This is a more challenging task, as the variable measured with error enters not only in the kernel function but also in the polynomial approximation.…”

Consistency and asymptotic normality for a nonparametric prediction under measurement errors

“…For density estimators see, inter alia, Zhang (1990), Fan (1991), Fan (1992), Masry (1993), Fan and Liu (1997), Cator (2001), van Es and Uh (2004) and van Es and Uh (2005). For regression, see Fan et al (1990), Fan and Masry (1992), Delaigle et al (2009) and Honda (2010). In addition, nonparametric deconvolution estimation of density and regression under heterogeneous measurement errors has been considered by Meister (2007, 2008) and Meister (2009).…”

Nonparametric regression for dependent data in the errors-in-variables problem

“…When we carry out nonparametric estimation of regression functions, one of the most familiar estimators has been the local constant type estimator of Fan and Truong (1993). The estimator is based on the idea of the deconvolution kernel density estimator and is very recently extended to the higher order versions in Delaigle et al (2009). There are some other kinds of nonparametric regression estimators such as those of Schennach (2004), Comte and Taupin (2007), and Hu and Schennach (2008).…”

“…There are some other kinds of nonparametric regression estimators such as those of Schennach (2004), Comte and Taupin (2007), and Hu and Schennach (2008). Hereafter we concentrate on deconvolution kernel estimators of Fan and Truong (1993) and Delaigle et al (2009).…”

“…In this paper, we examine how the LRD of (Y i , X i ) affects asymptotic properties of the local constant and linear estimators of Fan and Truong (1993) and Delaigle et al (2009). We also follow Wu and Mielniczuk (2002) and use the results of Giraitis et al (1996) as in Kulik (2008).…”