## The Annals of Statistics

### Nonasymptotic bounds for autoregressive time series modeling

#### Abstract

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.

#### Article information

Source
Ann. Statist., Volume 29, Number 2 (2001), 417-444.

Dates
First available in Project Euclid: 24 December 2001

https://projecteuclid.org/euclid.aos/1009210547

Digital Object Identifier
doi:10.1214/aos/1009210547

Mathematical Reviews number (MathSciNet)
MR1863964

Zentralblatt MATH identifier
1041.62074

#### Citation

Goldenshluger, Alexander; Zeevi, Assaf. Nonasymptotic bounds for autoregressive time series modeling. Ann. Statist. 29 (2001), no. 2, 417--444. doi:10.1214/aos/1009210547. https://projecteuclid.org/euclid.aos/1009210547

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