Revisiting Révész's stochastic approximation method for the estimation of a regression function

Details Revisiting Révész's stochastic approximation method for the estimation of a regression function Revisiting Révész's stochastic approximation method for the estimation of a regression function

2008-12-20

arxiv.org/abs/0812.3973v1

Mathematics

Statistics Theory

Scientific paper

In a pioneer work, R\'ev\'esz (1973) introduces the stochastic approximation method to build up a recursive kernel estimator of the regression function $x\mapsto E(Y|X=x)$. However, according to R\'ev\'esz (1977), his estimator has two main drawbacks: on the one hand, its convergence rate is smaller than that of the nonrecursive Nadaraya-Watson's kernel regression estimator, and, on the other hand, the required assumptions on the density of the random variable $X$ are stronger than those usually needed in the framework of regression estimation. We first come back on the study of the convergence rate of R\'ev\'esz's estimator. An approach in the proofs completely different from that used in R\'ev\'esz (1977) allows us to show that R\'ev\'esz's recursive estimator may reach the same optimal convergence rate as Nadaraya-Watson's estimator, but the required assumptions on the density of $X$ remain stronger than the usual ones, and this is inherent to the definition of R\'ev\'esz's estimator. To overcome this drawback, we introduce the averaging principle of stochastic approximation algorithms to construct the averaged R\'ev\'esz's regression estimator, and give its asymptotic behaviour. Our assumptions on the density of $X$ are then usual in the framework of regression estimation. We prove that the averaged R\'ev\'esz's regression estimator may reach the same optimal convergence rate as Nadaraya-Watson's estimator. Moreover, we show that, according to the estimation by confidence intervals point of view, it is better to use the averaged R\'ev\'esz's estimator rather than Nadaraya-Watson's estimator.

Affiliated with

Also associated with

No associations

Rating

Revisiting Révész's stochastic approximation method for the estimation of a regression function does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Revisiting Révész's stochastic approximation method for the estimation of a regression function, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Revisiting Révész's stochastic approximation method for the estimation of a regression function will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-103894