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Biometrika 1976 63(2):367-380; doi:10.1093/biomet/63.2.367
© 1976 by Biometrika Trust
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On the Cramér-Wold factorization

E. J. GODOLPHIN

Department of Statistics and Computer Science, Royal Holloway College Egham, Surrey

Several authors have considered the problem of deriving the q moving average parameters of an autoregressive-moving average process of order (p, q), given estimates of the p auto-regressive parameters and the first q members of the parent autocorrelation function; the problem was first raised by H. Cramér and H. Wold. The problem is discussed in connexion with the parameter estimation procedures of Walker (1961, 1962) and of Box & Jenkins (1970). An approach to the problem is described which is most suitable in the common cases where q is small; otherwise a factorization routine due to Wold (1954) is found to be preferable for most practical purposes when q is moderately large, whilst an iterative algorithm due to Tunnicliffe Wilson (1969) has more desirable properties in all other cases.

Key Words: Autocorrelation estimation • Autoregressive-moving average process • Polynomial factorization • Time series estimation


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