On the asymptotic behavior of the Nadaraya-Watson estimator associated with the recursive SIR method

TL;DR

This paper studies the asymptotic properties of a recursive Nadaraya-Watson estimator within a semiparametric regression model, combining recursive sliced inverse regression for parameter estimation with a recursive kernel estimator for the regression function.

Contribution

It introduces a recursive approach to estimate both the model parameter and the regression function, establishing their almost sure convergence and asymptotic normality.

Findings

Proves almost sure convergence of the recursive estimator.

Establishes asymptotic normality of the estimator.

Demonstrates effectiveness through simulations.

Abstract

We investigate the asymptotic behavior of the Nadaraya-Watson estimator for the estimation of the regression function in a semiparametric regression model. On the one hand, we make use of the recursive version of the sliced inverse regression method for the estimation of the unknown parameter of the model. On the other hand, we implement a recursive Nadaraya-Watson procedure for the estimation of the regression function which takes into account the previous estimation of the parameter of the semiparametric regression model. We establish the almost sure convergence as well as the asymptotic normality for our Nadaraya-Watson estimator. We also illustrate our semiparametric estimation procedure on simulated data.