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January, 1980 Asymptotically Efficient Selection of the Order of the Model for Estimating Parameters of a Linear Process
Ritei Shibata
Ann. Statist. 8(1): 147-164 (January, 1980). DOI: 10.1214/aos/1176344897

Abstract

Let $\{x_t\}$ be a linear stationary process of the form $x_t + \Sigma_{1\leqslant i<\infty}a_ix_{t-i} = e_t$, where $\{e_t\}$ is a sequence of i.i.d. normal random variables with mean 0 and variance $\sigma^2$. Given observations $x_1, \cdots, x_n$, least squares estimates $\hat{a}(k)$ of $a' = (a_1, a_2, \cdots)$, and $\hat{\sigma}^2_k$ of $\sigma^2$ are obtained if the $k$th order autoregressive model is assumed. By using $\hat{a}(k)$, we can also estimate coefficients of the best predictor based on $k$ successive realizations. An asymptotic lower bound is obtained for the mean squared error of the estimated predictor when $k$ is selected from the data. If $k$ is selected so as to minimize $S_n(k) = (n + 2k)\hat{\sigma}^2_k$, then the bound is attained in the limit. The key assumption is that the order of the autoregression of $\{x_t\}$ is infinite.

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Ritei Shibata. "Asymptotically Efficient Selection of the Order of the Model for Estimating Parameters of a Linear Process." Ann. Statist. 8 (1) 147 - 164, January, 1980. https://doi.org/10.1214/aos/1176344897

Information

Published: January, 1980
First available in Project Euclid: 12 April 2007

zbMATH: 0425.62069
MathSciNet: MR557560
Digital Object Identifier: 10.1214/aos/1176344897

Subjects:
Primary: 62M10
Secondary: 62E20 , 62M20

Keywords: Autoregression , efficiency , Model selection , prediction , time series models

Rights: Copyright © 1980 Institute of Mathematical Statistics

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Vol.8 • No. 1 • January, 1980
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