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

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  • Karim Benhenni
  • Mustapha Rachdi
  • Yingcai Su

Abstract

This article considers estimation of regression function $$f$$ in the fixed design model $$Y(x_i)=f(x_i)+ \epsilon (x_i), i=1,\ldots ,n$$ , by use of the Gasser and Müller kernel estimator. The point set $$\{ x_i\}_{i=1}^{n}\subset [0,1]$$ constitutes the sampling design points, and $$\epsilon (x_i)$$ are correlated errors. The error process $$\epsilon $$ is assumed to satisfy certain regularity conditions, namely, it has exactly $$k$$ ( $$=\!0, 1, 2, \ldots $$ ) 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 $$(2k+1)$$ th mixed partial derivatives of the autocovariance function along the diagonal of the unit square. Simulation results for the model of $$k$$ 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. Copyright Springer-Verlag Berlin Heidelberg 2013

Suggested Citation

  • Karim Benhenni & Mustapha Rachdi & Yingcai Su, 2013. "The effect of the regularity of the error process on the performance of kernel regression estimators," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 76(6), pages 765-781, August.
  • Handle: RePEc:spr:metrik:v:76:y:2013:i:6:p:765-781
    DOI: 10.1007/s00184-012-0414-8
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    References listed on IDEAS

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    1. Vicente Núñez-Antón & Juan Rodríguez-Póo & Philippe Vieu, 1999. "Longitudinal data with nonstationary errors: a nonparametric three-stage approach," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 8(1), pages 201-231, June.
    2. Su, Yingcai & Cambanis, Stamatis, 1993. "Sampling designs for estimation of a random process," Stochastic Processes and their Applications, Elsevier, vol. 46(1), pages 47-89, May.
    3. Ferreira, Eva & Núñez-Antón, Vicente & Rodríguez-Póo, Juan, 1997. "Kernel regression estimates of growth curves using nonstationary correlated errors," Statistics & Probability Letters, Elsevier, vol. 34(4), pages 413-423, June.
    4. Blanke, Delphine & Vial, Céline, 2008. "Assessing the number of mean square derivatives of a Gaussian process," Stochastic Processes and their Applications, Elsevier, vol. 118(10), pages 1852-1869, October.
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