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ARMA processes have maximal entropy among time series with prescribed autocovariances and impulse responses

Published online by Cambridge University Press:  01 July 2016

Jürgen Franke
Jürgen Franke*
Affiliation:
University of Frankfurt
*
∗Postal address: Department of Mathematics, University of Frankfurt, P.O. Box 111 932, D-6000 Frankfurt, West Germany.
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Abstract

The maximum-entropy approach to the estimation of the spectral density of a time series has become quite popular during the last decade. It is closely related to the fact that an autoregressive process of order p has maximal entropy among all time series sharing the same autocovariances up to lag p. We give a natural generalization of this result by proving that a mixed autoregressive-moving-average process (ARMA process) of order (p, q) has maximal entropy among all time series sharing the same autocovariances up to lag p and the same impulse response coefficients up to lag q. The latter may be estimated from a finite record of the time series, for example by using a method proposed by Bhansali (1976). By the way, we give a result on the existence of ARMA processes with prescribed autocovariances up to lag p and impulse response coefficients up to lag q.

Keywords

MIXED AUTOREGRESSIVE-MOVING-AVERAGE PROCESSMAXIMUM ENTROPY METHOD
Type
Research Article
Information
Advances in Applied Probability , Volume 17 , Issue 4 , December 1985 , pp. 810 - 840
Copyright
Copyright © Applied Probability Trust 1985 

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