Local and Global Asymptotic Inferences for the Smoothing Spline Estimate By

This article presents the first comprehensive studies on the local and global inferences for the smoothing spline estimate in a unified asymptotic framework. The novel functional Bahadur representation is developed as the theoretical foundation of this article, and is also of independent interest. Based on that, we establish four interconnected inference procedures: (i) Point-wise Confidence Interval; (ii) Local Likelihood Ratio Testing; (iii) Simultaneous Confidence Band (SCB); (iv) Global Likelihood Ratio Testing. In particular, our C.I. is proven to be asymptotically valid at any point over the support , and is extraordinarily shorter than the classical Bayesian C.I. (Wahba, 1983). We also unveil new Wilk's phenomena arising from the local/global likelihood ratio testing, and further show that the global testing is more powerful/efficient than the local one in terms of the smaller minimum separation rate. It is also worthy noting that our SCB is the first one applicable to the general quasi-likelihood models. Furthermore, the inference optimality/efficiency issues are carefully addressed. As a by-product of this article, we discover some surprising asymptotic equivalence phenomenon between the periodic and non-periodic smoothing splines in terms of inferences. 1. Introduction. Smoothing spline models provide a very general framework for data analysis, modeling and learning in a variety of fields; see [57, 58, 21]. As far as we are aware, the existing literature are mostly concerned about the global convergence properties or methodological studies of smoothing spline estimate. Unfortunately, a systematic and rigorous theoretical study on their asymptotic inferences is almost nonexistent. This is partly due to the technical restrictions of the widely used equivalent kernel method. The novel Functional Bahadur Representation (FBR) we develop brings several major breakthroughs into the inference studies. The main purpose of this paper is to propose a series of local and global inference procedures for a univariate smooth curve based on FBR as the theoretical foundation. Moreover, we carefully investigate the inference optimality/efficiency that has not been well treated in the smoothing spline literature. In this paper, we consider a general class of nonparametric regression models that covers the least square regression and logistic regression. The equivalent kernel method has long been used as a standard tool in dealing with the asymptotics of the smoothing splines, but it is only restricted to the simple least square regression; see [48, 38]. Moreover, this classical method only " approximates " the reproducing kernel function and the approximation formula becomes extremely complicated …