FPCA-based estimation for generalized functional partially linear models
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FPCA-based estimation for generalized functional partially linear models Ruiyuan Cao1 · Jiang Du1 · Jianjun Zhou2 · Tianfa Xie1 Received: 11 April 2018 / Revised: 6 September 2018 © Springer-Verlag GmbH Germany, part of Springer Nature 2019
Abstract In real data analysis, practitioners frequently come across the case that a discrete response will be related to both a function-valued random variable and a vector-value random variable as the predictor variables. In this paper, we consider the generalized functional partially linear models (GFPLM). The infinite slope function in the GFPLM is estimated by the principal component basis function approximations. Then, we consider the theoretical properties of the estimator obtained by maximizing the quasi likelihood function. The asymptotic normality of the estimator of the finite dimensional parameter and the rate of convergence of the estimator of the infinite dimensional slope function are established, respectively. We investigate the finite sample properties of the estimation procedure via Monte Carlo simulation studies and a real data analysis. Keywords Generalized linear model · Functional partially linear model · Quasi likelihood · Karhunen–Loève representation Mathematics Subject Classification 62G08 · 62G20
Du’s work is supported by the National Natural Science Foundation of China (Nos. 11501018, 11771032), the Science and Technology Project of Beijing Municipal Education Commission (KM201910005015) and Program for Rixin Talents in Beijing University of Technology. Cao’s work is supported by the National Natural Science Foundation of China (No. 11701020). Zhou’s work is supported by the National Natural Science Foundation of China (No. 11861074). Xie’s work is supported by the National Natural Science Foundation of China (No. 11571340) and the Science and Technology Project of Beijing Municipal Education Commission (KM201710005032).
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Jiang Du [email protected]
1
College of Applied Sciences, Beijing University of Technology, Beijing 100124, China
2
School of Mathematics and Statistics, Yunnan University, Kunming 650091, China
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R. Cao et al.
1 Introduction To model various continuous and discrete data, generalized linear models (GLMs) are proposed by Nelder and Wedderburn (1972). GLMs are advantageous for analyzing the relationship between the predictors and the function of the mean of continuous or discrete response. For a detail review of GLMs, we refer the readers to McCullagh and Nelder (1989) and the references therein. In recent years, functional data analysis (FDA) has received substantial attentions in various applied fields, including genetics, brain imaging, biomedical studies, environmental sciences, public health, signal processing, chemometrics, and so on. Recently, many researchers concentrated on developing statistical inference methods via combining the nonparametric methods with FDA, for example, Aneiros-Pérez and Vieu (2006), Zhou and Chen (2012), Peng et al. (2016), Yu et al. (2018) and among others. For more details on FDA, we re
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