Summary
Two sets of modified kernel estimates of a regression function are proposed: one when a bound on the regression function is known and the other when nothing of this sort is at hand. Explicit bounds on the mean square errors of the estimators are obtained. Pointwise as well as uniform consistency in mean square and consistency in probability of the estimators are proved. Speed of convergence in each case is investigated.
Similar content being viewed by others
References
Bierens, H. J. (1983). Uniform consistency of kernel estmators of a regression function under generalized conditions,J. Amer. Statist. Ass.,78, 699–707.
Devroye, L. P. and Wagner, T. (1980). Distribution free consistency results in nonparametric discrimination and regression function estimations,Ann. Statist.,8, 231–239.
Johns, M. V. and Van Ryzin, J. (1972). Convergence rates for empirical Bayes two-action problems II. Continuous case,Ann. Math. Statist.,43, 934–947.
Kale, B. (1962). A note on a problem in estimation,Biometrika 49, 553–556.
Menon, V. V., Prasad, B. and Singh, R. S. (1984). Nonparametric recursive estimates of a probability density function and its derivations,J. Statist. Plann. Inference,9, 73–82.
Nadaraya, E. A. (1965). On nonparametric estimates of density function and regression curves.Theor. Prob. Appl.,10, 186–190.
Noda, K. (1976). Estimation of a regression function by the Parzen kernel-type density estimators,Ann. Inst. Statist. Math.,28, 221–234.
Parzen, E. (1962). On estimation of a probability density function and mode,Ann. Math. Statist.,33, 1065–1076.
Rosenblatt, M. (1956). Remarks on some nonparametric estimators of density function,Ann. Math. Statist.,27, 832–837.
Schuster, E. F. (1972). Joint asymptotic distribution of the estimated regression function at a finite number of distinct points,Ann. Math. Statist.,43, 84–88.
Schuster, E. F. and Yakowitz, S. (1979). Contributions to the theory of nonparametric regression, with application to system identification,Ann. Statist.,7, 139–149.
Singh, R. S. (1977a). Improvement on some known nonparametric uniformly consistent estimators of derivatives of a density,Ann. Statist.,5, 394–399.
Singh, R. S. (1977b). Applications of estimators of a density and its derivatives to certain statistical problems,J. R. Statist. Soc.,39, 357–363.
Singh, R. S. and Tracy, D. S. (1977). Strongly consistent estimators, ofk-th order regression curves and rates of convergence,Zeit. Wahrscheinlichkeitsth.,40, 339–348.
Singh, R. S. (1979). Mean, square errors of estimates of a density and its derivatives,Biometrika,66, 177–180.
Singh, R.S. (1980). Estimation of regression curves when the conditional density of the predictor variable is in scale exponential family,Multivar. Statist. Anal. (ed. R. P. Gupta).
Singh, R. S. (1981a). On the exact asymptotic behaviour of estimators of a density and its derivatives,Ann. Statist.,9, 453–456.
Singh, R. S. (1981b). Speed of convergence in nonparametric estimation of a multivariate μ-density and its mixed partial derivatives,J. Statist. Plann. Inference,5, 287–298.
Singh, R. S. and Ullah, A. (1984). Nonparametric recursive estimation of a multivariate, marginal and conditional DGP with an application to specification of econometric models, to appear inCommun. Statist.
Singh, R. S. and Ullah, A. (1985). Nonparametric time series estimation of joint DGP, conditional DGP and vector autoregression,Econ. Theory,1, 27–52.
Spiegelman, C. and Sacks, J. (1980). Consistent, window estimation in nonparametric regression,Ann. Statist.,33, 1065–1076.
Author information
Authors and Affiliations
Additional information
Major work of this research was completed during the first author's two visits (November–December, 1983 and August–September 1984) to the second author at the Universite du Quebec a Montreal. Part of the work of the second author was supported by the Air Force Office of Scientific Research under contract F49620-85-C-0008 while he was at the University of Pittsburgh during Spring in 1985.
About this article
Cite this article
Singh, R.S., Ahmad, M. Modified nonparametric kernel estimates of a regression function and their consistencies with rates. Ann Inst Stat Math 39, 549–562 (1987). https://doi.org/10.1007/BF02491489
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF02491489