×

A note on strong convergence rates in nonparametric regression. (English) Zbl 0835.62046

Summary: The strong convergence rates in nonparametric regression estimation have been mostly discussed when the error variables in the regression models have finite variances. A few recent studies concern heavy-tailed error distributions for two comparable methods using the kernel and the \(k\)- nearest neighbor estimators. The obtained convergence rates are however noncomparable. Assuming the error variables have finite \(p\)th moments for the same \(p\), \(1< p< 2\), we derive comparable strong convergence rates for these two estimators via a unified approach. This improves the existing results for both the kernel estimator and the \(k\)-nearest neighbor estimator.

MSC:

62G20 Asymptotic properties of nonparametric inference
62G07 Density estimation
Full Text: DOI

References:

[1] Benedetti, J. O., On the nonparametric estimation of regression functions, J. Roy. Statist. Soc. Ser. B., 39, 699-707 (1977)
[2] Brunk, H. D., Estimation of isotonic regression, (Puri, M. L., Nonparametric Techniques in Statistical Inference (1970), Cambridge Univ. Press: Cambridge Univ. Press Cambridge), 177-195 · Zbl 0030.20003
[3] Cheng, K. F.; Lin, P. E., Nonparametric estimation of a regression function, Z. Wahrsch. verw. Gebiete, 57, 223-233 (1981) · Zbl 0443.62029
[4] Cheng, P. E., Strong consistency of nearest neighbor regression function estimators, J. Multivariate Anal., 15, 63-72 (1984) · Zbl 0542.62031
[5] Clark, R. M., Nonparametric estimation of a smooth regression function, J. Roy. Statitst. Soc. Ser. B, 39, 107-113 (1977) · Zbl 0355.62036
[6] Devroye, L. P.; Wagner, T. J., The strong uniform consistency of nearest neighbor density estimates, Ann. Statist., 5, 536-540 (1977) · Zbl 0367.62061
[7] Fan, J., Local linear regression smoothers and their minimax efficiencies, Ann. Statist., 21, 196-216 (1993) · Zbl 0773.62029
[8] Feller, W., (An Introduction to Probability Theory and its Applications, Vol. 1 (1968), Wiley: Wiley New York) · Zbl 0155.23101
[9] Gasser, Th; Müller, H.-G, Kernel estimation of regression functions, (Gasser, Th; Rosenblatt, M., Smoothing Techniques for Curve Estimation (1979), Springer: Springer Heidelberg), 23-68 · Zbl 0418.62033
[10] Hall, P., On iterated logarithm laws for linear arrays and nonparametric regression estimators, Ann. Probab., 19, 740-757 (1991) · Zbl 0733.60047
[11] Härdle, W.; Janssen, P.; Serfling, R., Strong uniform consistency rates for estimators of conditional functionals, Ann. Statist., 16, 1428-1449 (1988) · Zbl 0672.62050
[12] Krzyżak, A.; Pawlak, M., The pointwise rate of convergence of the kernel regression estimate, J. Statist. Plann. Inference, 16, 159-166 (1987) · Zbl 0616.62050
[13] Loève, M., (Probability Theory, 1 (1977), Springer: Springer New York) · Zbl 0359.60001
[14] Mack, Y. P.; Rosenblatt, M., Multivariate \(k\)-nearest neighbor density estimates, J. Multivariate Anal., 9, 1-15 (1979) · Zbl 0406.62023
[15] Mack, Y. P., Local properties of the \(k\)-NN regression estimates, SIAM J. Algebraic Discrete Methods, 2, 311-323 (1981) · Zbl 0499.62037
[16] Mandelbrot, B., The Pareto-Lévy law and the distribution of income, Internat. Econom. Rev., 1, 79-106 (1960) · Zbl 0201.51101
[17] Mukerjee, H., Nearest neighbor regression with heavy-tailed errors, Ann, Statist., 21, 681-693 (1993) · Zbl 0779.62036
[18] Müller, H.-G; Stadtmüller, U., Estimation of heteroscedasticity in regression analysis, Ann. Statist., 15, 610-625 (1987) · Zbl 0632.62040
[19] Nadaraya, E. A., On estimating regression, Theory Probab. Appl., 9, 141-142 (1964) · Zbl 0136.40902
[20] Prakasa Rao, B. L.S, Nonparametric Function Estimation (1983), Academic Press: Academic Press New York · Zbl 0542.62025
[21] Priestley, M. B.; Chao, M. T., Non-parametric function fitting, J. Roy. Statist. Soc. Ser. B, 34, 385-392 (1972) · Zbl 0263.62044
[22] Rice, J., Bandwidth choice for nonparametric regression, Ann. Statist., 12, 1215-1230 (1984) · Zbl 0554.62035
[23] Royall, R. M., A class of nonparametric estimates of a smooth regression function, (Ph.D. Dissertation (1966), Stanford University) · Zbl 0197.16303
[24] Stone, C. J., Consistent nonparametric regression, Ann. Statist., 5, 595-645 (1977) · Zbl 0366.62051
[25] Watson, G. S., Smooth regression analysis, Sankhyā Ser. A, 26, 359-372 (1964) · Zbl 0137.13002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.