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November, 1976 The Convergence of Some Recursions
E. J. Hannan
Ann. Statist. 4(6): 1258-1270 (November, 1976). DOI: 10.1214/aos/1176343658


In connection with a range of stationary time series models, particularly ARMAX models, recursive calculations of the parameter vector seem important. In these the estimate, $\theta(n)$, from observations to time $n$, is calculated as $\theta(n) = \theta(n - 1) + k_n$ where $k_n$ depends only on $\theta(n - 1), \theta(n - 2), \cdots$ and the data to time $n$. The convergence of two recursions is proved for the simple model $x(n) = \varepsilon(n) + \alpha\varepsilon(n - 1), |\alpha| < 1$, where the $\varepsilon(n)$ are stationary ergodic martingale differences with $E\{\varepsilon(n)^2\mid\mathscr{F}_{n-1}\} = \sigma^2$. The method of proof consists in reducing the study of the recursion to that of a recursion involving the data only through the $\theta(n)$. It seems that many of the recursions introduced for ARMAX models may be treated in this way and the nature of the extensions of the theory is discussed.


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E. J. Hannan. "The Convergence of Some Recursions." Ann. Statist. 4 (6) 1258 - 1270, November, 1976.


Published: November, 1976
First available in Project Euclid: 12 April 2007

zbMATH: 0336.62076
MathSciNet: MR519092
Digital Object Identifier: 10.1214/aos/1176343658

Primary: 62M10
Secondary: 62L12

Keywords: ARMA models , martingale , Recursive calculation , stochastic approximation

Rights: Copyright © 1976 Institute of Mathematical Statistics

Vol.4 • No. 6 • November, 1976
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