The Annals of Statistics

Estimation for Autoregressive Moving Average Models in the Time and Frequency Domains

T. W. Anderson

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The autoregressive moving average model is a stationary stochastic process $\{y_t\}$ satisfying $\sum^p_{k=0} \beta_ky_{t-k} = \sum^q_{g=0} \alpha_g\nu_{t-g}$, where the (unobservable) process $\{v_t\}$ consists of independently identically distributed random variables. The coefficients in this equation and the variance of $v_t$ are to be estimated from an observed sequence $y_1, \cdots, y_T$. To apply the method of maximum likelihood under normality the model is modified (i) by setting $y_0 = y_{-1} = \cdots = y_{1-p} = 0$ and $\nu_0 = v_{-1} = \cdots = v_{1-q} = 0$ and alternatively (ii) by setting $y_0 \equiv y_T, \cdots, y_{1-p} \equiv y_{T+1-p}$ and $v_0 \equiv v_T, \cdots, v_{1-q} \equiv v_{T+1-q}$; the former lead to procedures in the time domain and the latter to procedures in the frequency domain. Matrix methods are used for a unified development of the Newton-Raphson and scoring iterative procedures; most of the procedures have been obtained previously by different methods. Estimation of the covariances of the moving average part is also treated.

Article information

Ann. Statist., Volume 5, Number 5 (1977), 842-865.

First available in Project Euclid: 12 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 62M10: Time series, auto-correlation, regression, etc. [See also 91B84]
Secondary: 62H99: None of the above, but in this section

Maximum likelihood estimation autoregressive moving average models Newton-Raphson and scoring iterative procedures time and frequency domains time series analysis


Anderson, T. W. Estimation for Autoregressive Moving Average Models in the Time and Frequency Domains. Ann. Statist. 5 (1977), no. 5, 842--865. doi:10.1214/aos/1176343942.

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