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Biometrika 1984 71(1):207-211; doi:10.1093/biomet/71.1.207
© 1984 by Biometrika Trust
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MISCELLANEA

Duality and other properties of multiplicative seasonal autoregressive-moving average models

A. I. McLEOD

Department of Statistical and Actuarial Sciences, University of Western Ontario London, Ontario, Canada

Model duality is defined between four models referred to as the primal, the dual, the autoregressive adjoint and the moving average adjoint. A duality theorem which generalizes the results of Box & Pierce (1970) and Pierce (1970) is presented. Applications of this duality theorem to autoregressive-moving average models and multiplicative seasonal autoregressive-moving average models are discussed. These applications include:

(i) a convenient method for calculating the covariance matrix of the estimated

parameters;

(ii) convenient formulae for the variances of the residual autocorrelations; (iii) the distribution of the inverse partial autocorrelations.

Finally, a useful approximation to the covariance determinant of multiplicative seasonal models is derived.

Key Words: Covariance determinant • Inverse partial autocorrelations • Parameter estimation • Residual autocorrelation


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