The Annals of Statistics

Local and global asymptotic inference in smoothing spline models

Zuofeng Shang and Guang Cheng

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Abstract

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.

Article information

Source
Ann. Statist., Volume 41, Number 5 (2013), 2608-2638.

Dates
First available in Project Euclid: 19 November 2013

Permanent link to this document
https://projecteuclid.org/euclid.aos/1384871347

Digital Object Identifier
doi:10.1214/13-AOS1164

Mathematical Reviews number (MathSciNet)
MR3161439

Zentralblatt MATH identifier
1293.62107

Subjects
Primary: 62G20: Asymptotic properties 62F25: Tolerance and confidence regions
Secondary: 62F15: Bayesian inference 62F12: Asymptotic properties of estimators

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

Citation

Shang, Zuofeng; Cheng, Guang. Local and global asymptotic inference in smoothing spline models. Ann. Statist. 41 (2013), no. 5, 2608--2638. doi:10.1214/13-AOS1164. https://projecteuclid.org/euclid.aos/1384871347


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Supplemental materials

  • Supplementary material: Supplement to “Local and global asymptotic inference in smoothing spline models”. The supplementary materials contain all the proofs of the theoretical results in the present paper.