Additive functional regression in reproducing kernel Hilbert spaces under smoothness condition
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Additive functional regression in reproducing kernel Hilbert spaces under smoothness condition Yuzhu Tian1 · Hongmei Lin2
· Heng Lian3 · Zengyan Fan4
Received: 5 December 2019 / Accepted: 22 September 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract Additive functional model is one popular semiparametric approach for regression with a functional predictor. Optimal prediction error rate has been demonstrated in the framework of reproducing kernel Hilbert spaces (RKHS), which only depends on the property of the RKHS but not on the smoothness of the function. We extend this previous theoretical result by establishing faster convergence rates under stronger conditions which is reduced to existing results when the stronger condition is removed. In particular, our result shows that with a smoother function the convergence rate of the estimator is faster. Keywords Convergence rate · Functional data · Reproducing kernel Hilbert space
1 Introduction For the past several decades, mathematical analysis of data sets involving observations that are curves has become a significant branch of statistics called functional data analysis and has received a lot of attention. As an initial approach, classical linear models were first extended to deal with such data and is well-summarized in the monograph (Ramsay and Silverman 2005). Nonparametric approaches to functional regression have also been developed in Ferraty and Vieu (2002). Other works on functional data analysis include Cardot et al. (2003), Cai and Hall (2006), Preda
B
Hongmei Lin [email protected]
1
School of Mathematics and Statistics, Henan University of Science and Technology, LuoYang, China
2
School of Statistics and Information, Shanghai University of International Business and Economics, Shanghai, China
3
Department of Mathematics, City University of Hong Kong, Hong Kong, China
4
School of Science and Technology, Singapore University of Social Sciences, Singapore, Singapore
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Y. Tian et al.
(2007), Lian (2007), Ait-Saidi et al. (2008), Yao et al. (2005), Crambes et al. (2009), Ferraty et al. (2011) and Lian (2011), among many others. When the response Y is a scalar, the functional linear model is given by Y =a+
1
X (t) f (t)dt + ,
(1)
0
where is a mean zero error with finite variance, a ∈ R is the intercept and f ∈ L 2 ([0, 1]) is the slope function to be estimated. There are many different extensions of the model (1) to incorporate some nonlinear effects of the functional predictor. One can use fully nonparametric approaches as in Ferraty and Vieu (2003), or semiparametric models including the single-index models (Ait-Saidi et al. 2008; Chen et al. 2011) and the additive models (Müller and Yao 2008; Zhu et al. 2014). In this paper, we revisit the additive functional model, which originates from McLean et al. (2014) and Müller et al. (2013) who investigate its computational and theoretical properties. Wang and Ruppert (2015) put the model over a reproducing kernel Hilbert space and established its minimax
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