  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.
