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On the asymptotic normality of kernel regression estimators of the mode in the nonparametric random design model. (English) Zbl 1041.62033

Summary: In the nonparametric regression model with random design, where the regression function \(m\) is given by \(m(x)={\mathbb E}(Y\mid X=x)\), estimation of the location \(\theta\) (mode) of a unique maximum of \(m\) by the location \(\widehat{\theta}\) of a maximum of the Nadaraya-Watson kernel estimator \(\widehat m\) for the curve \(m\) is considered. Within this setting, we obtain consistency and asymptotic normality results for \(\widehat{\theta}\) under mild local smoothness assumptions on \(m\) and the design density \(g\) of \(X\). The estimation of the size of the maximum is also considered as well as the estimation of a unique zero of the regression function. As a by-product, we obtain an asymptotic normality result for the Nadaraya-Watson estimator itself which improves on previous results.

MSC:

62G08 Nonparametric regression and quantile regression
62G20 Asymptotic properties of nonparametric inference
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