Abstract
In the case of the random design nonparametric regression, one recursive local polynomial smoother is considered. Expressions for the bias and the variance matrix of the estimators of the regression function and its derivatives are obtained under dependence conditions (strongly mixing processes). The obtained Mean Squared Error is shown to be larger than those of the analogous nonrecursive regression estimators, although retaining the same convergence rate. The properties of strong consistency with convergence rates are established for the proposed estimators. Finally, in order to analyse the influence of both the sample size and the dependence in the behaviour of the proposed recursive estimator, a simulation study is performed.
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References
Cleveland, W.S. (1979). Robust locally weighted regression and smoothing scatterplots.Journal of the American Statistical Association,74, 829–836.
Deheuvels, P. (1974). Conditions nécessaires et sufissantes de convergence ponetuelle presque sûre et uniforme presque sûre des estimatours de la densité.C. R. Acad. Sci. Paris. Sér. A.,278, 1217–1220.
Devroye, L. and T.J. Wagner (1980). On theL 1 convergence of kernel estimators of regression function with application in discrimination.Z. Wahrsch. Verw. Gebiete,51, 15–25.
Fan, J. (1992). Design-adaptive nonparametric regression.Journal of the American Statistical Association,87, 998–1004.
Fan, J. (1993). Local lincar regression smoothers and their minimax efficiency.Annals of Statistics,21, 196–216.
Fan, J. and I. Gijbels (1992). Variable bandwith and local linear regression smoothers.Annals of Statistics,20, 2008–2036.
Fan, J. and I. Gijbels (1995). Data-driven bandwidth selection in local polynomial fitting: variable bandwidth and spatial adaption.Journal of the Royal Statistical Society, B,57, 2, 371–394.
Fan, J. and I. Gijbels (1996).Local Polynomial Fitting and its Applications. Chapman and Hall, London.
Greblicky, W. and M. Pawlak (1987). Necessary and sufficient consistency conditions for a recursive kernel regression estimate.Journal of the Multivariate Analisys,23, 67–76.
Hastic, T. and C. Loader (1993). Local regression: automatic kernel carpentry.Statistical Science,8, 2, 120–143.
Krzyzak, A. and M. Pawlak (1984). Almost everywhere convergence of recursive regression function estimate and classification.IEEE Transactions in Information Therapy,30, 91–93.
Lejeune, M. (1985). Estimation non-paramétrique par noyaux: régression polynomiale mobile.Revue de Statist. Appliq.,33, 43–68.
Masry, E. (1987). Almost sure convergence of recursive density estimators for stationary mixing processes.Statistics and Probability Letters,5, 249–254.
Masry, E. (1996a). Multivariate regression estimation: local polynomial fitting for time series.Stochastic Processes and Applications,65, 81–101.
Masry, E. (1996b). Multivariate local polynomial regression for time serics: uniform strong consistency and rates.Journal of Time Series Analisys,17, 571–599.
Masry, E. and J. Fan (1997). Local polynomial estimation of regression function for mixing processes.Scandinavian Journal of Statistics,24, 165–179.
McLeish, D.L. (1975). A maximal inequality and dependent strong law.Annals of Probability,3, 829–839.
Revesz, P. (1977). How to apply the method of stochastic approximation in the nonparametric estimation of a regression function.Mathematische Operationsforschung, Serie Statistics,8, 119–126.
Roussas, G.G. (1992). Exact rates of almost sure convergence of a recursive kernel estimate of a probability density function: application to regression and hazard rate estimation.Nonparametric Statistics,1, 171–195.
Roussas, G.G. and L.T. Tran (1992). Asymptotic normality of the recursive kernel regression estimate under dependence conditions.Annals of Statistics,20, 1 98–120.
Ruppert, D. and P. Wand (1994). Multivariate locally weighted least squares regression.Annals of Statistics,22, 1346–1370.
Ruppert, D., S.J. Sheather and M.P. Wand (1995). An effective bandwidth selector for local least squares regression.Journal of the American Statistical Association,90, 1257–1270.
Stone, C.J. (1977). Consistent nonparametric regression.Annals of Statistics,5, 595–620.
Tsybakov, A.B. (1990). Locally-polynomial algorithms of passive stochastic approximation.Problems of Control and Information Theory,19, 3, 181–195.
Vilar, J.M. and Vilar, J.A. (1993). Estimacion no paramétrica, recursiva, de curvas.Estadística Española,35, 134, 579–616.
Vilar, J.A. and Vilar, J.M. (1998). Recursive estimation of regression functions by local polynomial fitting.Annals of the Institute of Statistical Mathematics,50, 4, 729–754.
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This work has been partially supported by DGES Grant PB98-0182-C02-01 and by the Xunta de Galicia Grant XUGA10501B97
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Vilar-Fernández, J.M., Vilar-Fernández, J.A. Recursive local polynomial regression under dependence conditions. Test 9, 209–232 (2000). https://doi.org/10.1007/BF02595859
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DOI: https://doi.org/10.1007/BF02595859