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April 2001 Nonasymptotic bounds for autoregressive time series modeling
Alexander Goldenshluger, Assaf Zeevi
Ann. Statist. 29(2): 417-444 (April 2001). DOI: 10.1214/aos/1009210547


The subject of this paper is autoregressive (AR) modeling of a stationary, Gaussian discrete time process, based on a finite sequence of observations. The process is assumed to admit an AR($\infty$) representation with exponentially decaying coefficients. We adopt the nonparametric minimax framework and study how well the process can be approximated by a finite­order AR model. A lower bound on the accuracy of AR approximations is derived, and a nonasymptotic upper bound on the accuracy of the regularized least squares estimator is established. It is shown that with a “proper” choice of the model order, this estimator is minimax optimal in order. These considerations lead also to a nonasymptotic upper bound on the mean squared error of the associated one­step predictor. A numerical study compares the common model selection procedures to the minimax optimal order choice.


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Alexander Goldenshluger. Assaf Zeevi. "Nonasymptotic bounds for autoregressive time series modeling." Ann. Statist. 29 (2) 417 - 444, April 2001.


Published: April 2001
First available in Project Euclid: 24 December 2001

zbMATH: 1041.62074
MathSciNet: MR1863964
Digital Object Identifier: 10.1214/aos/1009210547

Primary: 62G05 , 62M10 , 62M20

Keywords: autoregressive approximation , minimax risk , rates of convergence

Rights: Copyright © 2001 Institute of Mathematical Statistics


Vol.29 • No. 2 • April 2001
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