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LOCAL AND GLOBAL ASYMPTOTIC INFERENCE FOR THE SMOOTHING SPLINE ESTIMATE

This article presents methods for local and global inference for the smoothing spline estimation in a unified asymptotic framework. A functional Bahadur representation is developed as the theoretical foundation of this article, and is also of independent interest. Based on that, we establish four inter-connected procedures for inference: (i) Point-wise Confidence Interval (CI); (ii) Local Likelihood Ratio Testing; (iii) Simultaneous Confidence Band (SCB); (iv) Global Likelihood Ratio Testing. In particular, our CI is proved to be asymptotically valid at any point in the support, and is shorter than the classical Bayesian CI (Wahba, 1983) on average. We also discuss a version of Wilks’ phenomenon arising from local/global likelihood ratio testing, and further show that global testing is more sensitive than local testing in terms of the minimax rate of testing defined by Ingster (1993). It is also worth noting that our SCB is the first one applicable to the 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 non-periodic smoothing splines in terms of inference.