Annals of Statistics
- Ann. Statist.
- Volume 43, Number 4 (2015), 1742-1773.
Nonparametric inference in generalized functional linear models
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.
Ann. Statist., Volume 43, Number 4 (2015), 1742-1773.
Received: May 2014
Revised: February 2015
First available in Project Euclid: 17 June 2015
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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
- Supplement to “Nonparametric inference in generalized functional linear models”. Proofs are provided.