Strong Consistency of Kernel Regression Estimate

Affiliation(s)

Department of Statistics and Finance, University of Science and Technology of China, Hefei, China.

Department of Statistics and Finance, University of Science and Technology of China, Hefei, China.

ABSTRACT

In this paper, regression function estimation from independent and
identically distributed data is considered. We establish strong
pointwise consistency of the famous Nadaraya-Watson estimator under weaker
conditions which permit to apply kernels with unbounded support and even not
integrable ones and provide a general approach for constructing strongly
consistent kernel estimates of regression functions.

Cite this paper

W. Cui and M. Wei, "Strong Consistency of Kernel Regression Estimate,"*Open Journal of Statistics*, Vol. 3 No. 3, 2013, pp. 179-182. doi: 10.4236/ojs.2013.33020.

W. Cui and M. Wei, "Strong Consistency of Kernel Regression Estimate,"

References

[1] E. A. Nadaraya, “On Estimating Regression,” Theory of Probability and Its Applications, Vol. 9, No. 1, 1964, pp. 141-142. doi:10.1137/1109020

[2] G. S. Watson, “Smooth Regression Analysis,” Sankhya: The Indian Journal of Statistics, Series A, Vol. 26, No. 4, 1964, pp. 359-372.

[3] C. J. Stone, “Consistent Nonparametric Regression,” Annals of Statistics, Vol. 5, No. 4, 1977, pp. 595-620. doi:10.1214/aos/1176343886

[4] E. F. Schuster and S. Yakowitz, “Contributions to the Theory of Nonparametric Regression, with Application to System Identification,” Annals Statistics, Vol. 7, No. 1, 1979, pp. 139-149. doi:10.1214/aos/1176344560

[5] T. Gasser and H. G. Müller, “Estimating Regression Functions and Their Derivatives by the Kernel Method,” Scandinavian Journal of Statistics, Vol. 11, 1984, pp. 171-185.

[6] Y. P. Mack and H. G. Müller, “Derivative Estimation in Nonparametric Regression with Random Predictor,” Sankhya: The Indian Journal of Statistics (Series A), Vol. 51, No. 1, 1989, pp. 59-72.

[7] W. Greblicki and M. Pawlak, “Cascade Non-Linear System Identification by a Non-Parametric Method,” International Journal of Systems Science, Vol. 25, No. 1, 1994, 129-153. doi:10.1080/00207729408928949

[8] M. Kohler, A. Krzy?ak and H. Walk, “Strong Consisten cy of Automatic Kernel Regression Estimates,” Annals of the Institute of Statistical Mathematics, Vol. 55, No. 2, 2003, pp. 287-308.

[9] M. Kohler, A. Krzy?ak and H. Walk, “Rates of Convergence for Partition-Zing and Nearest Neighbor Regression Estimates with Unbounded Data,” Journal of Multivariate Analysis, Vol. 97, No. 2, 2006, pp. 311-323. doi:10.1016/j.jmva.2005.03.006

[10] H. Walk, “Strong Universal Consistency of Smooth Kernel Regression Estimates,” Annals of the Institute of Statistical Mathematics, Vol. 57, No. 4, 2005, pp. 665-685. doi:10.1007/BF02915432

[11] L. Devroye, “On the Almost Everywhere Convergence of Nonparametric Regression Function Estimates,” Annals Statistics, Vol. 9, No. 6, 1981, pp. 1310-1319. doi:10.1214/aos/1176345647

[12] L. C. Zhao and Z. B. Fang, “Strong Convergence of Kernel Estimates of Nonparametric Regression Functions,” Chinese Annals of Mathematics, Series B, Vol. 6, No. 2, 1985, pp. 147-155.

[13] W. Greblicki, A. Krzyzak and M. Pawlak, “Distribution Free Pointwise Consistency of Kernel Regression Estimate,” Annals Statistics, Vol. 12, No. 4, 1984, pp. 1570-1575. doi:10.1214/aos/1176346815

[1] E. A. Nadaraya, “On Estimating Regression,” Theory of Probability and Its Applications, Vol. 9, No. 1, 1964, pp. 141-142. doi:10.1137/1109020

[2] G. S. Watson, “Smooth Regression Analysis,” Sankhya: The Indian Journal of Statistics, Series A, Vol. 26, No. 4, 1964, pp. 359-372.

[3] C. J. Stone, “Consistent Nonparametric Regression,” Annals of Statistics, Vol. 5, No. 4, 1977, pp. 595-620. doi:10.1214/aos/1176343886

[4] E. F. Schuster and S. Yakowitz, “Contributions to the Theory of Nonparametric Regression, with Application to System Identification,” Annals Statistics, Vol. 7, No. 1, 1979, pp. 139-149. doi:10.1214/aos/1176344560

[5] T. Gasser and H. G. Müller, “Estimating Regression Functions and Their Derivatives by the Kernel Method,” Scandinavian Journal of Statistics, Vol. 11, 1984, pp. 171-185.

[6] Y. P. Mack and H. G. Müller, “Derivative Estimation in Nonparametric Regression with Random Predictor,” Sankhya: The Indian Journal of Statistics (Series A), Vol. 51, No. 1, 1989, pp. 59-72.

[7] W. Greblicki and M. Pawlak, “Cascade Non-Linear System Identification by a Non-Parametric Method,” International Journal of Systems Science, Vol. 25, No. 1, 1994, 129-153. doi:10.1080/00207729408928949

[8] M. Kohler, A. Krzy?ak and H. Walk, “Strong Consisten cy of Automatic Kernel Regression Estimates,” Annals of the Institute of Statistical Mathematics, Vol. 55, No. 2, 2003, pp. 287-308.

[9] M. Kohler, A. Krzy?ak and H. Walk, “Rates of Convergence for Partition-Zing and Nearest Neighbor Regression Estimates with Unbounded Data,” Journal of Multivariate Analysis, Vol. 97, No. 2, 2006, pp. 311-323. doi:10.1016/j.jmva.2005.03.006

[10] H. Walk, “Strong Universal Consistency of Smooth Kernel Regression Estimates,” Annals of the Institute of Statistical Mathematics, Vol. 57, No. 4, 2005, pp. 665-685. doi:10.1007/BF02915432

[11] L. Devroye, “On the Almost Everywhere Convergence of Nonparametric Regression Function Estimates,” Annals Statistics, Vol. 9, No. 6, 1981, pp. 1310-1319. doi:10.1214/aos/1176345647

[12] L. C. Zhao and Z. B. Fang, “Strong Convergence of Kernel Estimates of Nonparametric Regression Functions,” Chinese Annals of Mathematics, Series B, Vol. 6, No. 2, 1985, pp. 147-155.

[13] W. Greblicki, A. Krzyzak and M. Pawlak, “Distribution Free Pointwise Consistency of Kernel Regression Estimate,” Annals Statistics, Vol. 12, No. 4, 1984, pp. 1570-1575. doi:10.1214/aos/1176346815