Rates of strong uniform consistency for local least squares kernel regression estimators

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 Einmahl and Mason [2000. An empirical process approach to the uniform consistency of kernel-type function estimators. J. Theoret. Probab. 13.1, 1–37.] and Deheuvels and Mason [2004. General asymptotic confidence bands based on kernel-type function estimators. Statist. Infer. Stochastic Process. 7.3, pp. 225–277] relative to uniform deviations of nonparametric kernel estimators.