The Annals of Statistics

Variable Bandwidth Kernel Estimators of Regression Curves

Hans-Georg Muller and Ulrich Stadtmuller

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In the model $Y_i = g(t_i) + \varepsilon_i,\quad i = 1,\cdots, n,$ where $Y_i$ are given observations, $\varepsilon_i$ i.i.d. noise variables and $t_i$ nonrandom design points, kernel estimators for the regression function $g(t)$ with variable bandwidth (smoothing parameter) depending on $t$ are proposed. It is shown that in terms of asymptotic integrated mean squared error, kernel estimators with such a local bandwidth choice are superior to the ordinary kernel estimators with global bandwidth choice if optimal bandwidths are used. This superiority is maintained in a certain sense if optimal local bandwidths are estimated in a consistent manner from the data, which is proved by a tightness argument. The finite sample behavior of a specific local bandwidth selection procedure based on the Rice criterion for global bandwidth choice [Rice (1984)] is investigated by simulation.

Article information

Ann. Statist., Volume 15, Number 1 (1987), 182-201.

First available in Project Euclid: 12 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 62G05: Estimation
Secondary: 65J02 65D10: Smoothing, curve fitting

Nonparametric kernel regression consistent bandwidth choice Rice criterion local bandwidths asymptotic optimality tightness in $C$ Gaussian limiting process


Muller, Hans-Georg; Stadtmuller, Ulrich. Variable Bandwidth Kernel Estimators of Regression Curves. Ann. Statist. 15 (1987), no. 1, 182--201. doi:10.1214/aos/1176350260.

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