The Annals of Statistics

Recursive Estimation Based on ARMA Models

E. J. Hannan

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A recursive estimate of the stochastic structure of a stationary time series is constructed based on the assumption that the true structure is ARMA, i.e., has a rational spectrum. The estimate is recursive in the sense that each successive estimate is obtained from the previous one by a relatively simple adjustment, that could be effected in a "real time" situation. The procedure is basically that of updating a regression when all variates involved are constructed from previous estimates of the parameter vector. The strong convergence of the estimate to the true value is established as well as a result relating to the rate of convergence.

Article information

Ann. Statist., Volume 8, Number 4 (1980), 762-777.

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: 62N15 62L12: Sequential estimation

ARMA models recursive estimation strong convergence martingales real time calculation


Hannan, E. J. Recursive Estimation Based on ARMA Models. Ann. Statist. 8 (1980), no. 4, 762--777. doi:10.1214/aos/1176345069.

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