Local and global asymptotic inference in smoothing spline models

TL;DR

This paper develops a unified asymptotic framework for local and global inference in smoothing spline models, introducing a new functional Bahadur representation and various inference procedures with improved confidence intervals.

Contribution

It introduces a novel functional Bahadur representation and develops interconnected inference procedures, including confidence intervals and likelihood ratio tests, for smoothing spline models.

Findings

Confidence intervals are asymptotically valid and shorter than Bayesian ones.

First simultaneous confidence bands applicable to general quasi-likelihood models.

Discovery of a relationship between periodic and nonperiodic smoothing splines in inference.

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.