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September, 1977 Estimation for Autoregressive Moving Average Models in the Time and Frequency Domains
T. W. Anderson
Ann. Statist. 5(5): 842-865 (September, 1977). DOI: 10.1214/aos/1176343942


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.


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T. W. Anderson. "Estimation for Autoregressive Moving Average Models in the Time and Frequency Domains." Ann. Statist. 5 (5) 842 - 865, September, 1977.


Published: September, 1977
First available in Project Euclid: 12 April 2007

zbMATH: 0368.62075
MathSciNet: MR448762
Digital Object Identifier: 10.1214/aos/1176343942

Primary: 62M10
Secondary: 62H99

Keywords: autoregressive moving average models , maximum likelihood estimation , Newton-Raphson and scoring iterative procedures , time and frequency domains , time series analysis

Rights: Copyright © 1977 Institute of Mathematical Statistics


Vol.5 • No. 5 • September, 1977
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