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Rates of strong uniform consistency for local least squares kernel regression estimators. (English) Zbl 1128.62046

Summary: We establish exact rates of strong uniform consistency for the multivariate Nadaraya-Watson kernel estimator of the regression function and its derivatives. As a special case, we treat the local linear estimator of the regression and the local polynomial smoothers of derivatives of the regression in the more convenient univariate setting. Our methods of proofs are based upon modern empirical process theory in the spirit of the results of U. Einmahl and D M. Mason [An empirical process approach to the uniform consistency of kernel-type function estimators. J. Theor. Probab. 13, No. 1, 1–37 (2000; Zbl 0995.62042)] and P. Deheuvels and D. M. Mason [General asymptotic confidence bands based on kernel-type function estimators. Stat. Inf. Stoch. Process. 7, No. 3, 225–277 (2004; Zbl 1125.62314)] relative to uniform deviations of nonparametric kernel estimators.

MSC:

62G08 Nonparametric regression and quantile regression
62G20 Asymptotic properties of nonparametric inference
60F15 Strong limit theorems
62H12 Estimation in multivariate analysis

Software:

KernSmooth
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References:

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