Nonasymptotic bounds for autoregressive time series modeling

Research output: Contribution to journalArticlepeer-review


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 in AR(∞) representation with exponentially decaying coefficients. We adopt the nonparametric mini max framework and study how well the process can be approximated by a finite-order 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 one-step predictor. A numerical study compares the common model selection procedures to the minimax optimal order choice.

Original languageEnglish
Pages (from-to)417-444
Number of pages28
JournalAnnals of Statistics
Issue number2
StatePublished - Apr 2001


  • Autoregressive approximation
  • Minimax risk
  • Rates of convergence

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty


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