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2017 Kernel estimates of nonparametric functional autoregression models and their bootstrap approximation
Tingyi Zhu, Dimitris N. Politis
Electron. J. Statist. 11(2): 2876-2906 (2017). DOI: 10.1214/17-EJS1303


This paper considers a nonparametric functional autoregression model of order one. Existing contributions addressing the problem of functional time series prediction have focused on the linear model and literatures are rather lacking in the context of nonlinear functional time series. In our nonparametric setting, we define the functional version of kernel estimator for the autoregressive operator and develop its asymptotic theory under the assumption of a strong mixing condition on the sample. The results are general in the sense that high-order autoregression can be naturally written as a first-order AR model. In addition, a component-wise bootstrap procedure is proposed that can be used for estimating the distribution of the kernel estimation and its asymptotic validity is theoretically justified. The bootstrap procedure is implemented to construct prediction regions that achieve good coverage rate. A supporting simulation study is presented in the end to illustrate the theoretical advances in the paper.


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Tingyi Zhu. Dimitris N. Politis. "Kernel estimates of nonparametric functional autoregression models and their bootstrap approximation." Electron. J. Statist. 11 (2) 2876 - 2906, 2017.


Received: 1 October 2016; Published: 2017
First available in Project Euclid: 8 August 2017

zbMATH: 1373.62170
MathSciNet: MR3694571
Digital Object Identifier: 10.1214/17-EJS1303

Primary: 37M10 , 60G25 , 62G08 , 62G09

Keywords: $\alpha$-mixing , functional time series , Nonparametric autoregression , prediction region , regression-type bootstrap


Vol.11 • No. 2 • 2017
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