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.
Keywords
MathematicsAutoregressive modelCombinatoricsSigmaOrder (exchange)Upper and lower boundsStatisticsSequence (biology)Mathematical analysis
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Publication Info
- Year
- 1980
- Type
- article
- Volume
- 8
- Issue
- 1
- Citations
- 555
- Access
- Closed
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Cite This
Ritei Shibata
(1980).
Asymptotically Efficient Selection of the Order of the Model for Estimating Parameters of a Linear Process.
The Annals of Statistics
, 8
(1)
.
https://doi.org/10.1214/aos/1176344897
Identifiers
- DOI
- 10.1214/aos/1176344897