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November, 1975 Maximum Likelihood Estimation of Parameters of Autoregressive Processes with Moving Average Residuals and Other Covariance Matrices with Linear Structure
T. W. Anderson
Ann. Statist. 3(6): 1283-1304 (November, 1975). DOI: 10.1214/aos/1176343285

Abstract

The autoregressive process with moving average residuals is a stationary process $\{y_t\}$ satisfying $\sum^p_{s = 0} \beta_sy_{t - s} = \sum^q_{j = 0} \alpha_j\nu_{t - j}$, where the sequence $\{\nu_t\}$ consists of independently identically distributed (unobservable) random variables. The distribution of $y_1,\cdots, y_T$ can be approximated by the distribution of the $T$-component vector $\mathbf{y}$ satisfying $\sum^p_{s = 0} \beta_s\mathbf{K}_s\mathbf{y} = \sum^q_{j = 0} \alpha_j\mathbf{J}_j\mathbf{v}$, where $\mathbf{v}$ has covariance matrix $\sigma^2\mathbf{I}, \mathbf{K}_s = \mathbf{J}_s = \mathbf{L}^s$, and $\mathbf{L}$ is the $T \times T$ matrix with 1's immediately below the main diagonal and 0's elsewhere. Maximum likelihood estimates are obtained when $\mathbf{v}$ has a normal distribution. The method of scoring is used to find estimates defined by linear equations which are consistent, asymptotically normal, and asymptotically efficient (as $T\rightarrow \infty$). Several special cases are treated. It is shown how to calculate the estimates.

Citation

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T. W. Anderson. "Maximum Likelihood Estimation of Parameters of Autoregressive Processes with Moving Average Residuals and Other Covariance Matrices with Linear Structure." Ann. Statist. 3 (6) 1283 - 1304, November, 1975. https://doi.org/10.1214/aos/1176343285

Information

Published: November, 1975
First available in Project Euclid: 12 April 2007

zbMATH: 0331.62067
MathSciNet: MR383672
Digital Object Identifier: 10.1214/aos/1176343285

Subjects:
Primary: 62M10
Secondary: 62H99

Rights: Copyright © 1975 Institute of Mathematical Statistics

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Vol.3 • No. 6 • November, 1975
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