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October 2013 Local and global asymptotic inference in smoothing spline models
Zuofeng Shang, Guang Cheng
Ann. Statist. 41(5): 2608-2638 (October 2013). DOI: 10.1214/13-AOS1164


This article studies local and global inference for smoothing spline estimation in a unified asymptotic framework. We first introduce a new technical tool called functional Bahadur representation, which significantly generalizes the traditional Bahadur representation in parametric models, that is, Bahadur [Ann. Inst. Statist. Math. 37 (1966) 577–580]. Equipped with this tool, we develop four interconnected procedures for inference: (i) pointwise confidence interval; (ii) local likelihood ratio testing; (iii) simultaneous confidence band; (iv) global likelihood ratio testing. In particular, our confidence intervals are proved to be asymptotically valid at any point in the support, and they are shorter on average than the Bayesian confidence intervals proposed by Wahba [J. R. Stat. Soc. Ser. B Stat. Methodol. 45 (1983) 133–150] and Nychka [J. Amer. Statist. Assoc. 83 (1988) 1134–1143]. We also discuss a version of the Wilks phenomenon arising from local/global likelihood ratio testing. It is also worth noting that our simultaneous confidence bands are the first ones applicable to general quasi-likelihood models. Furthermore, issues relating to optimality and efficiency are carefully addressed. As a by-product, we discover a surprising relationship between periodic and nonperiodic smoothing splines in terms of inference.


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Zuofeng Shang. Guang Cheng. "Local and global asymptotic inference in smoothing spline models." Ann. Statist. 41 (5) 2608 - 2638, October 2013.


Published: October 2013
First available in Project Euclid: 19 November 2013

zbMATH: 1293.62107
MathSciNet: MR3161439
Digital Object Identifier: 10.1214/13-AOS1164

Primary: 62F25 , 62G20
Secondary: 62F12 , 62F15

Keywords: asymptotic normality , functional Bahadur representation , local/global likelihood ratio test , simultaneous confidence band , smoothing spline

Rights: Copyright © 2013 Institute of Mathematical Statistics


Vol.41 • No. 5 • October 2013
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