NONPARAMETRIC INFERENCE IN GENERALIZED FUNCTIONAL LINEAR MODELS

• 成果类型: Article
• 署名作者: Shang, Zuofeng; Cheng, Guang
• 署名单位: Purdue University System; Purdue University
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/15-AOS1322
• 发表日期: 2015
• 页码: 1742-1773
• 关键词: integrodifferential equations; exponential-families; regression problems; correlated errors; longitudinal data; likelihood ratio; greens-function; CONVERGENCE; prediction; Minimax

摘要

We propose a roughness regularization approach in making nonparametric inference for generalized functional linear models. In a reproducing kernel Hilbert space framework, we construct asymptotically valid confidence intervals for regression mean, prediction intervals for future response and various statistical procedures for hypothesis testing. In particular, one procedure for testing global behaviors of the slope function is adaptive to the smoothness of the slope function and to the structure of the predictors. As a by-product, a new type of Wilks phenomenon [Ann. Math. Stat. 9 (1938) 60-62; Ann. Statist. 29 (2001) 153-193] is discovered when testing the functional linear models. Despite the generality, our inference procedures are easy to implement. Numerical examples are provided to demonstrate the empirical advantages over the competing methods. A collection of technical tools such as integro-differential equation techniques [Trans. Amer Math. Soc. (1927) 29 755-800; Trans. Amer. Math. Soc. (1928) 30 453-471; Trans. Amer Math. Soc. (1930) 32 860-868], Stein's method [Ann. Statist. 41 (2013) 2786- 2819] [Stein, Approximate Computation of Expectations (1986) IMS] and functional Bahadur representation [Ann. Statist. 41 (2013) 2608-2638] are employed in this paper.