## The Annals of Statistics

- Ann. Statist.
- Volume 8, Number 1 (1980), 147-164.

### Asymptotically Efficient Selection of the Order of the Model for Estimating Parameters of a Linear Process

#### 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.

#### Article information

**Source**

Ann. Statist., Volume 8, Number 1 (1980), 147-164.

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176344897

**Digital Object Identifier**

doi:10.1214/aos/1176344897

**Mathematical Reviews number (MathSciNet)**

MR557560

**Zentralblatt MATH identifier**

0425.62069

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62M10: Time series, auto-correlation, regression, etc. [See also 91B84]

Secondary: 62M20: Prediction [See also 60G25]; filtering [See also 60G35, 93E10, 93E11] 62E20: Asymptotic distribution theory

**Keywords**

Autoregression time series models prediction efficiency model selection

#### Citation

Shibata, Ritei. Asymptotically Efficient Selection of the Order of the Model for Estimating Parameters of a Linear Process. Ann. Statist. 8 (1980), no. 1, 147--164. doi:10.1214/aos/1176344897. https://projecteuclid.org/euclid.aos/1176344897