Annals of Statistics

Nonparametric inference in generalized functional linear models

Zuofeng Shang and Guang Cheng

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Abstract

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.

Article information

Source
Ann. Statist., Volume 43, Number 4 (2015), 1742-1773.

Dates
Received: May 2014
Revised: February 2015
First available in Project Euclid: 17 June 2015

Permanent link to this document
https://projecteuclid.org/euclid.aos/1434546221

Digital Object Identifier
doi:10.1214/15-AOS1322

Mathematical Reviews number (MathSciNet)
MR3357877

Zentralblatt MATH identifier
1317.62042

Subjects
Primary: 62G20: Asymptotic properties 62F25: Tolerance and confidence regions
Secondary: 62F15: Bayesian inference 62F12: Asymptotic properties of estimators

Keywords
Generalized functional linear models minimax adaptive test nonparametric inference reproducing kernel Hilbert space roughness regularization

Citation

Shang, Zuofeng; Cheng, Guang. Nonparametric inference in generalized functional linear models. Ann. Statist. 43 (2015), no. 4, 1742--1773. doi:10.1214/15-AOS1322. https://projecteuclid.org/euclid.aos/1434546221


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