  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.
