Abstract
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.
Keywords
MathematicsEstimatorBandwidth (computing)Kernel regressionKernel smootherApplied mathematicsKernel (algebra)SmoothingStatisticsMathematical optimizationKernel methodCombinatoricsComputer scienceArtificial intelligence
Related Publications
Locally Adaptive Bandwidth Choice for Kernel Regression Estimators
Abstract Kernel estimators with a global bandwidth are commonly used to estimate regression functions. On the other hand, it is obvious that the choice of a local bandwidth can ...
How Far are Automatically Chosen Regression Smoothing Parameters from their Optimum?
Abstract We address the problem of smoothing parameter selection for nonparametric curve estimators in the specific context of kernel regression estimation. Call the "optimal ba...
Smoothing Parameter Selection in Nonparametric Regression Using an Improved Akaike Information Criterion
Summary Many different methods have been proposed to construct nonparametric estimates of a smooth regression function, including local polynomial, (convolution) kernel and smoo...
The Differentiation of Pseudo-Inverses and Nonlinear Least Squares Problems Whose Variables Separate
For given data $(t_i ,y_i ),i = 1, \cdots ,m$, we consider the least squares fit of nonlinear models of the form \[ \eta ({\bf a},{\boldsymbol \alpha} ;t) = \sum _{j = 1}^n {a_j...
Spline Smoothing: The Equivalent Variable Kernel Method
The spline smoothing approach to nonparametric regression and curve estimation is considered. It is shown that, in a certain sense, spline smoothing corresponds approximately to...
Publication Info
- Year
- 1987
- Type
- article
- Volume
- 15
- Issue
- 1
- Citations
- 159
- Access
- Closed
External Links
Social Impact
Altmetric
PlumX Metrics
Social media, news, blog, policy document mentions
Citation Metrics
159
OpenAlex
Cite This
Hans‐Georg Müller,
Ulrich Stadtmüller
(1987).
Variable Bandwidth Kernel Estimators of Regression Curves.
The Annals of Statistics
, 15
(1)
.
https://doi.org/10.1214/aos/1176350260
Identifiers
- DOI
- 10.1214/aos/1176350260