ON NONPARAMETRIC KERNEL ESTIMATION OF THE MODE OF THE REGRESSION FUNCTION IN THE RANDOM DESIGN MODEL

Author: KLAUS ZIEGLER

Source: Journal of Nonparametric Statistics, Volume 14, Number 6, 2002 , pp. 749-774(26)

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Abstract:

In the nonparametric regression model with random design, where the regression function m is given by <$>m(x) = {open E}(Ymid X = x),<$> estimation of the location <$>theta<$> (mode) of a unique maximum of m by the location <$>hat{theta}<$> of a maximum of the Nadaraya-Watson kernel estimator <$>hat{m}<$> for the curve m is considered. Within this setting, we obtain consistency and asymptotic normality results for <$>hat{theta}<$> under very mild assumptions on m, the design density g of X and the kernel K. The bandwidths being considered in the present work are data-dependent of the type being generated by plug-in methods. The estimation of the size of the maximum is also considered as well as the estimation of a unique zero of the regression function. Applied to the estimation of the mode of a density, our methods yield some improvements on known results. As a by-product, we obtain some uniform consistency results for the (higher) derivatives of the Nadaraya-Watson estimator with a certain additional uniformity in the bandwiths. The proofs of those rely heavily on empirical process methods.

Keywords: Nonparametric regression; Random design; Mode; Kernel smoothing; Nadaraya-Watson estimator; Data-dependent bandwidths; Estimation of derivatives; Consistency, Asymptotic normality

Document Type: Research article

DOI: http://dx.doi.org/10.1080/10485250215321

Affiliations: 1: Mathematical Institute, University of Munich, Theresienstrasse 39, D-80333 Munich, Germany

Publication date: 2002-01-01