Nonasymptotic bounds for autoregressive time series modeling

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(co) representation with exponentiall y decaying coefficients. We adopt the nonparametric minimax framework and s tudy 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 nonasym ptotic upper bound on the accuracy of the regularized least squares estimat or is established. It is shown that with a "proper" choice of the model ord er, this estimator is minimax optimal in order. These considerations lead a lso to a nonasymptotic upper bound on the mean squared error of the associa ted one-step predictor, A numerical study compares the common model selecti on procedures to the minimax optimal order choice.