The effect of the regularity of the error process on the performance of kernel regression estimators

Abstract

This article considers estimation of regression function ff in the fixed design model Y(xi)=f(xi)+ϵ(xi),i=1,"¦,nY(xi)=f(xi)+ϵ(xi),i=1,"¦,n , by use of the Gasser and Müller kernel estimator. The point set {xi}ni=1⊂[0,1]{xi}i=1n⊂[0,1] constitutes the sampling design points, and ϵ(xi)ϵ(xi) are correlated errors. The error process ϵϵ is assumed to satisfy certain regularity conditions, namely, it has exactly kk ( =0,1,2,"¦=0,1,2,"¦ ) quadratic mean derivatives (q.m.d.). The quality of the estimation is measured by the mean squared error (MSE). Here the asymptotic results of the mean squared error are established. We found that the optimal bandwidth depends on the (2k+1)(2k+1) th mixed partial derivatives of the autocovariance function along the diagonal of the unit square. Simulation results for the model of kk th order integrated Brownian motion error are given in order to assess the effect of the regularity of this error process on the performance of the kernel estimator.

Department(s)

Mathematics

Document Type

Article

DOI

https://doi.org/10.1007/s00184-012-0414-8

Publication Date

2013

Journal Title

Metrika

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