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October 2003 Local asymptotics for polynomial spline regression
Jianhua Z. Huang
Ann. Statist. 31(5): 1600-1635 (October 2003). DOI: 10.1214/aos/1065705120


In this paper we develop a general theory of local asymptotics for least squares estimates over polynomial spline spaces in a regression problem. The polynomial spline spaces we consider include univariate splines, tensor product splines, and bivariate or multivariate splines on triangulations. We establish asymptotic normality of the estimate and study the magnitude of the bias due to spline approximation. The asymptotic normality holds uniformly over the points where the regression function is to be estimated and uniformly over a broad class of design densities, error distributions and regression functions. The bias is controlled by the minimum $L_\infty$ norm of the error when the target regression function is approximated by a function in the polynomial spline space that is used to define the estimate. The control of bias relies on the stability in $L_\infty$ norm of $L_2$ projections onto polynomial spline spaces. Asymptotic normality of least squares estimates over polynomial or trigonometric polynomial spaces is also treated by the general theory. In addition, a preliminary analysis of additive models is provided.


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Jianhua Z. Huang. "Local asymptotics for polynomial spline regression." Ann. Statist. 31 (5) 1600 - 1635, October 2003.


Published: October 2003
First available in Project Euclid: 9 October 2003

zbMATH: 1042.62035
MathSciNet: MR2012827
Digital Object Identifier: 10.1214/aos/1065705120

Primary: 62G07
Secondary: 62G20

Keywords: asymptotic normality , least squares , Nonparametric regression , polynomial regression , regression spline

Rights: Copyright © 2003 Institute of Mathematical Statistics


Vol.31 • No. 5 • October 2003
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