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april 1999 Random-design regression under long-range dependent errors
Sándor Csörgö, Jan Mielniczuk
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Bernoulli 5(2): 209-224 (april 1999).

Abstract

We consider the random-design nonparametric regression model with long-range dependent errors that may also depend on the independent and identically distributed explanatory variables. Disclosing a smoothing dichotomy, we show that the finite-dimensional distributions of the Nadaraya-Watson kernel estimator of the regression function converge either to those of a degenerate process with completely dependent marginals or to those of a Gaussian white-noise process. The first case occurs when the bandwidths are large enough in a specified sense to allow long-range dependence to prevail. The second case is for bandwidths that are small in the given sense, when both the required norming sequence and the limiting process are the same as if the errors were independent. The borderline situation results in a limiting convolution of the two cases. The main results contrast with previous findings for deterministic-design regression.

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Sándor Csörgö. Jan Mielniczuk. "Random-design regression under long-range dependent errors." Bernoulli 5 (2) 209 - 224, april 1999.

Information

Published: april 1999
First available in Project Euclid: 5 March 2007

zbMATH: 0946.62084
MathSciNet: MR1681695

Keywords: asymptotic finite-dimensional distributions , kernel estimators , long-range and short-range dependent errors , random-design regression , smoothing dichotomy

Rights: Copyright © 1999 Bernoulli Society for Mathematical Statistics and Probability

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Vol.5 • No. 2 • april 1999
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