Statistical inference for the functional quadratic quantile regression model
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Statistical inference for the functional quadratic quantile regression model Gongming Shi1 · Tianfa Xie1,2
· Zhongzhan Zhang1,2
Received: 29 January 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract In this paper, we develop statistical inference procedures for functional quadratic quantile regression model in which the response is a scalar and the predictor is a random function defined on a compact set of R. The functional coefficients are estimated by functional principal components. The asymptotic properties of the resulting estimators are established under mild conditions. In order to test the significance of the nonlinear term in the model, we propose a rank score test procedure. The asymptotic properties of the proposed test statistic are established. The proposed method provides a highly efficient and robust alternative to the least squares method, and can be conveniently implemented using existing R software package. Finally, we examine the performance of the proposed method for finite sample sizes by Monte Carlo simulation studies and illustrate it with a real data example. Keywords Quantile regression · Functional data · Functional quadratic regression · Rank score test Mathematics Subject Classification 62G08 · 62G20
Xie’s work is supported by the Science and Technology Project of Beijing Municipal Education Commission (KM201710005032, KM201910005015) and the National Natural Science Foundation of China (Nos. 11571340, 11971045). Zhang’s work is partly supported by the National Natural Science Foundation of China (Nos. 11271039, 11771032), and Education Ministry Funds for Doctor Supervisors (20131103110027).
B
Tianfa Xie [email protected]
1
College of Applied Sciences, Beijing University of Technology, Beijing 100124, China
2
Collaborative Innovation Center on Capital Social Construction and Social Management, Beijing University of Technology, Beijing 100124, China
123
G. Shi et al.
1 Introduction Functional linear regression models provide a useful framework for high dimensional data analysis, and they have recently attracted considerable attention due to a large variety of applications, see Cai and Hall (2006), Hall and Horowitz (2007), Cai and Yuan (2012), Kong et al. (2013) and among others. Let Y be a real-valued response variable and {X (t) : t ∈ [0, 1]} be a zero mean, second-order stochastic process with sample paths in L 2 ([0, 1]). To model the relationship between a functional predictor X (t) and the scalar response variable Y , Ramsay and Silverman (1997) introduced the following functional linear model: Y =μ+
1
X (t)β(t)dt + ε
(1)
0
where μ is a constant, denoting the intercept in the model and ε is the random error. There is a substantial literature on estimation of the unknown slope function β(·). Hall and Horowitz (2007) discussed the optimal convergence rates of the estimators based on functional principal components analysis (FPCA). Cardot et al. (2003) developed an estimator based on a B-splines expansion of the functional slope coefficient
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