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ARMA Processes Have Maximal Entropy among Time Series with Prescribed Autocovariances and Impulse Responses

Jürgen Franke
Advances in Applied Probability
Vol. 17, No. 4 (Dec., 1985), pp. 810-840
DOI: 10.2307/1427089
Stable URL: http://www.jstor.org/stable/1427089
Page Count: 31
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ARMA Processes Have Maximal Entropy among Time Series with Prescribed Autocovariances and Impulse Responses
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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.

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