Abstract
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.
Citation
E. J. Hannan. "The Convergence of Some Recursions." Ann. Statist. 4 (6) 1258 - 1270, November, 1976. https://doi.org/10.1214/aos/1176343658
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