Nonparametric inference in generalized functional linear models

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.

Published at http://dx.doi.org/10.1214/15-AOS1322 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

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• Prediction in functional linear regression
• Methodology and convergence rates for functional linear regression
• Gaussian approximations and multiplier bootstrap for maxima of sums of high-dimensional random vectors
• A reproducing kernel Hilbert space approach to functional linear regression
• Smoothing splines estimators for functional linear regression
• Local and global asymptotic inference in smoothing spline models

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