Reweighted Nadaraya-Watson Estimator of the Regression Mean

Raid B Salha,Hazem I El Shekh Ahmed

doi:10.5539/ijsp.v4n1p138

Abstract

In this paper, the estimation of the regression mean using the Reweighted Nadaraya-Watson (RNW) estimator has been considered. The RNW is a modification of the Nadaraya-Watson (NW) estimator in order to obtain some more refinement estimator. We have considered some conditions under which the asymptotic normality of the proposed estimator has been derived. Then we generalized this result to the multivariate case by considering the estimation of the regression mean at distinct points.

Highlights

Theory and methodology for nonparametric regression is well developed for the case of the estimation of the regression mean, to motivate the problem, consider a sequence of independent and identically distributed real random variables {(Xi, Yi)}in=1 with a joint pdf f(x, y) as a bivariate random variable (X, Y)
We have considered some conditions under which the asymptotic normality of the proposed estimator has been derived
The first result is stated in Theorem 1, where the asymptotic normality of the (RNW) estimator is shown

Summary

Introduction

Theory and methodology for nonparametric regression is well developed for the case of the estimation of the regression mean, to motivate the problem, consider a sequence of independent and identically distributed real random variables {(Xi, Yi)}in=1 with a joint pdf f(x, y) as a bivariate random variable (X, Y). The simple nonparametric regression function is written as. N are the corresponding responses, m(Xi) are the unknown regression mean functions to be estimated nonparametrically, and εi: i = 1,2, ... The regression mean function m(x) is the conditional mean, which is given by m(x). The regression mean function m(x) is estimated by m (x), where m x. Conditional density estimation was introduced by Rosenblatt (1969). Fan et al (1996) proposed a direct estimator based on a local polynomial estimation. One of the most widely used and studied estimators in the literature is the one proposed independently by Nadaraya (1964) and Watson (1964). Nadaraya–Watson kernel estimation is denoted by fNW(Y|x) and defined as, fNW(y|x) = ∑ni=1 Kh(y − Yi)wiNW(x)

Reweighted Nadaraya-Watson Estimator

Main Results

Conditions Consider the following conditions

SSE SSTO

Conclusion