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Nonparametric inference in generalized functional linear models. (English) Zbl 1317.62042
Summary: 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 [S. S. Wilks, Ann. Math. Stat. 9, 60–62 (1938; Zbl 0018.32003); J. Fan et al., ibid. 29, No. 1, 153–193 (2001; Zbl 1029.62042)] 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 [J. D. Tamarkin, Trans. Am. Math. Soc. 29, 755–800 (1927; JFM 53.0355.03); J. D. Tamarkin and R. E. Langer, Trans. Am. Math. Soc. 30, 453–471 (1928; JFM 54.0415.04); J. D. Tamarkin, Trans. Am. Math. Soc. 32, 860–868 (1930; JFM 56.1019.02)], Stein’s method [V. Chernozhukov et al., ibid. 41, No. 6, 2786–2819 (2013; Zbl 1292.62030); C. Stein, Approximate computation of expectations. Hayward, CA: Institute of Mathematical Statistics (1986; Zbl 0721.60016)] and functional Bahadur representation [the authors, ibid. 41, No. 5, 2608–2638 (2013; Zbl 1293.62107)] are employed in this paper.

##### MSC:
 62G20 Asymptotic properties of nonparametric inference 62F25 Parametric tolerance and confidence regions 62F15 Bayesian inference 62F12 Asymptotic properties of parametric estimators
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