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.