Abstract
Consider a regression model where the regression function is the sum of a linear and a nonparametric component. Assuming that the errors of the model follow a stationary strong mixing process with mean zero, the problem of bandwidth selection for a kernel estimator of the nonparametric component is addressed here. We obtain an asymptotic expression for an optimal band-width and we propose to use a plug-in methodology in order to estimate this bandwidth through preliminary estimates of the unknown quantities. Asymptotic optimality for the plug-in bandwidth is established.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aneiros-Pérez G (2002) On bandwidth selection in partial linear regression models under dependence. Statist. Probab. Lett.57, 393–401.
Aneiros-Pérez G, Quintela-del-Río A (2001) Modified Cross-Validation in Semiparametric Regression Models with Dependent Errors. Comm. Statist. Theory Methods30, 289–307
Beran J, Ghosh S (1998) Root-n-consistent Estimation in Partial Linear Models with Long-memory errors. Scand. J. Statist.25, 345–357
Brockwell P, Davis RA (1991) Time Series: Theory and Methods. Springer, New York
Cao R, García W, González W, Prada JM, Febrero M (1995) Predicting using Box-Jenkins, nonparametric, and bootstrap techniques. Technometrics37, 303–310
Chu CK (1989) Some results in nonparametric regression. Ph.D. Thesis, Department of Statistics, University of North Carolina, Chapel Hill
Doukhan P (1994) Mixing: properties and examples. Lecture Notes in Statistics,85, Springer-Verlag
Engle R, Granger C, Rice J, Weiss A (1986) Nonparametric estimates of the relation between weather and electricity sales. J. Amer. Statist. Assoc.81, 310–320.
Epanechnikov VA (1969) Nonparametric estimation of a multivariate probability density. Theory Probab. Appl.14, 153–158
Gao J (1995) Asymptotic theory for partly linear models. Comm. Statist. Theory Methods24, 1985–2009.
Gasser T, Müller HG (1984) Estimating regression functions and their derivatives by the kernel method. Scand. J. Statist.11, 171–185
Gasser T, Müller HG, Mammitzsch V (1985) Kernels for nonparametric curve estimation. J. Roy. Statist. Soc. Ser. B47, 238–252
Green P, Jennison C, Seheult A (1985) Analysis of field experiments by least squares smoothing. J. Roy. Statist. Soc. Ser. B47, 299–315
Györfi L, Härdle W, Sarda P, Vieu P (1990) Nonparametric curve estimation from time series. Lecture Notes in Statistics,60. Springer-Verlag
Härdle W (1990) Applied nonparametric regression. Cambridge University Press
Härdle W, Liang H, Gao, J (2000) Partially linear models. Contributions to Statistics. Physica-Verlag
Härdle W, Vieu P (1992) Kernel regression smoothing of time series. J. Time Ser. Anal.13, 209–232.
Hart J (1991) Kernel Regression Estimation with Time Series Errors. J. Roy. Statist. Soc. Ser. B53, 173–187
Heckman N (1986) Spline smoothing in partly linear model. J. Roy. Statist. Soc. Ser. B48, 244–248
Herrmann E, Gasser T, Kneip A (1992) Choice of bandwidth for kernel regression when residuals are correlated. Biometrika79, 783–795.
Kim TY (1994) Moment bounds for non-stationary dependent sequences. J. Appl. Probab.31, 731–742
Liang H, Härdle W, Werwatz A (1999) Asymptotic properties of nonparametric regression estimation in partly linear models. Discussion paper no. 55, Institut für Statistik und Ökonometrie, Humboldt-Universität zu Berlin.
Linton OB (1995) Second order approximation in the partially linear regression model. Econometrica63, 1079–1112
Park BU, Marron JS (1990) Comparison of Data-Driven Bandwidth Selectors. J. Amer. Statist. Assoc.85, 66–72
Quintela-del-Río A (1994) A plug-in technique in nonparametric regression with dependence. Comm. Statist. Theory Methods23, 2581–2603
Rice J (1986) Convergence rates for partially splined models. Statist. Probab. Lett.4, 203–208
Robinson P (1988) Root-N-consistent semiparametric regression. Econometrica56, 931–954
Roussas G, Ioannides DA (1988) Probability bounds for sums in triangular arrays of random variables under mixing conditions. Statist. Theor. and Dat. Anal. II, 293–308
Roussas G, Tran LT, Ioannides DA (1992) Fixed Design Regression for Time series: Asymptotic Normality. J. Multivariate Anal.40, 262–291
Sheather SJ, Jones MC (1991) A Reliable Data-based Bandwidth Selection Method for Kernel Density Estimation. J. Roy. Statist. Soc. Ser. B53, 683–690
Speckman P (1988) Kernel smoothing in partial linear models. J. Roy. Statist. Soc. Ser. B50, 413–436
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Aneiros-Pérez, G. Plug-in bandwidth choice for estimation of nonparametric part in partial linear regression models with strong mixing errors. Statistical Papers 45, 191–210 (2004). https://doi.org/10.1007/BF02777223
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02777223