Abstract
A somewhat more general class of nonparametric estimators for estimating an unknown regression functiong from noisy data is proposed. The regressor is assumed to be defined on the closed interval [0, 1]. This class of estimators is shown to be pointwisely consistent in the mean square sense and with probability one. Further, it turns out that these estimators can be applied to a wide class of noises.
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References
Ahmad, I. A. and Lin, P. E. (1984). Fitting a multiple regression function,J. Statist. Plann. Inference,9, 163–176.
Chen, K. F. and Lin, P. E. (1981). Nonparametric estimation of a regression function,Z. Wahrsch. Verw. Gebiete,57, 223–233.
Eubank, R. L. (1988).Spline Smoothing and Nonparametric Regression, Marcel Dekker, New York and Basel.
Galkowski, T. and Rutkowski, L. (1986). Nonparametric fitting of multivariate functions,IEEE Trans. Automat. Control,31, 785–787.
Gasser, T. and Müller, H. G. (1979). Kernel estimation of regression functions,Smoothing Techniques for Curve Estimation, (eds. T. Gasser and M. Rosenblatt),Lecture Notes in Math.,757, 26–68, Springer, Berlin-New York.
Gasser, T. and Müller, H. G. (1984). Estimating regression functions and their derivatives by the kernel method,Scand. J. Statist.,11, 171–185.
Georgiev, A. A. (1984a). Speed of convergence in nonparametric kernel estimation of a regression function and its derivatives,Ann. Inst. Statist. Math.,36, 455–462.
Georgiev, A. A. (1984b). On the recovery of functions and their derivatives from imperfect measurements,IEEE Trans. Systems Man Cybernet.,14, 900–903.
Georgiev, A. A. (1985). Nonparametric kernel algorithm for recovery of functions from noisy measurements with applications,IEEE Trans. Automat. Control,30, 782–784.
Georgiev, A. A. (1988). Consistent nonparametric multiple regression: The fixed design case,J. Multivariate Anal.,25, 100–110.
Georgiev, A. A. and Greblicki, W. (1986). Nonparametric function recovering from noisy observations,J. Statist. Plann. Inference,13, 1–14.
Priestley, M. B. and Chao, M. T. (1972). Nonparametric function fitting,J. Roy. Statist. Soc. Ser. B,34, 385–392.
Rice, J. and Rosenblatt, M. (1983). Smoothing splines: Regression, derivatives and deconvolution,Ann. Statist.,11, 141–156.
Rutkowski, L. and Rafajlowicz, E. (1989). On optimal global rate of convergence of some nonparametric identification procedures,IEEE Trans. Automat. Control,34, 1089–1091.
Stone, C. J. (1977). Consistent nonparametric regression,Ann. Statist.,5, 595–645.
Teicher, H. (1985). Almost certain convergence in double arrays,Z. Wahrsch. Verw. Gebiete,69, 331–345.
Walter, G. and Blum, J. (1979). Probability density estimation using delta sequences,Ann. Statist.,7, 328–340.
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Isogai, E. Nonparametric estimation of a regression function by delta sequences. Ann Inst Stat Math 42, 699–708 (1990). https://doi.org/10.1007/BF02481145
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DOI: https://doi.org/10.1007/BF02481145