Article,
Asymptotically efficient selection of the order of the model for estimating parameters of a linear process
The Annals of Statistics, 8 (1): 147--164 (January 1980)
DOI: 10.1214/aos/1176344897
Abstract
Let \textbackslash\x\_t\textbackslash\ be a linear stationary process of the form Math Processing Errorx\_t + \textbackslashSigma\_\1\textbackslashleqslant i\textless\textbackslashinfty\a\_ix\_\t-i\ = e\_t, where Math Processing Error\textbackslash\e\_t\textbackslash\ is a sequence of i.i.d. normal random variables with mean 0 and variance Math Processing Error\textbackslashsigmaˆ2. Given observations Math Processing Errorx\_1, \textbackslashcdots, x\_n, least squares estimates Math Processing Error\textbackslashhat\a\(k) of Math Processing Errora' = (a\_1, a\_2, \textbackslashcdots), and Math Processing Error\textbackslashhat\\textbackslashsigma\ˆ2\_k of Math Processing Error\textbackslashsigmaˆ2 are obtained if the Math Processing Errorkth order autoregressive model is assumed. By using Math Processing Error\textbackslashhat\a\(k), we can also estimate coefficients of the best predictor based on Math Processing Errork successive realizations. An asymptotic lower bound is obtained for the mean squared error of the estimated predictor when Math Processing Errork is selected from the data. If Math Processing Errork is selected so as to minimize Math Processing ErrorS\_n(k) = (n + 2k)\textbackslashhat\\textbackslashsigma\ˆ2\_k, then the bound is attained in the limit. The key assumption is that the order of the autoregression of Math Processing Error\textbackslash\x\_t\textbackslash\ is infinite.
