Adaptive kernel estimation of the mode in a nonparametric random design regression model

• Autorzy: Ziegler K.
• Języki publikacji: EN

Abstrakty

EN

In a nonparametric regression model with random design, where the regression function m is given by m (x) = E(Y |X = x), estimation of the location θ (mode) and size m (θ) of a unique maximum of m is considered. As estimators, location θ and size m (θ) of a maximum of the Nadaraya-Watson kernel estimator m for the curve m are chosen. Within this setting, we establish joint asymptotic normality and asymptotic independence for θ and m (θ) (which can be exploited for constructing simultaneous confidence intervals for θ and m (θ)) under mild local smoothness assumptions on m and the design density g (imposed in a neighborhood of θ). The bandwidths employed for m are data-dependent and of plug-in type. This is handled by viewing the estimators as stochastic processes indexed by a so-called scaling parameter and proving functional central limit theorems for those processes. In the same way, we obtain, as a by-product, an asymptotic normality result for the Nadaraya-Watson estymator itself at a finite number of distinct points, which improves on previous results.

Słowa kluczowe

EN

nonparametric regression; random design; mode; kernel smoothing; Nadaraya-Watson estimator; weak convergence; functional central limit theorems

• Wydawca: Wrocław University of Science and Technology
• Czasopismo: Probability and Mathematical Statistics
• Rocznik: 2004
• Tom: Vol. 24, Fasc. 2
• Strony: 213--235
• Opis fizyczny: Bibliogr. 32 poz.

Twórcy

autor

Ziegler K.

• Technical University of Ilmenau, Institute for Mathematics, Postfach 100565, D-98984 Ilmenau, Germany

Bibliografia

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