Unwin, Judith Mary (1984)
Covariance matrix theorems for estimators of time series models: With applications to active tracking problems.
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In this thesis, covariance matrices and generalised variances for maximum likelihood estimators of Gaussian autoregressive moving average time series models are derived. It is shown that estimators for pure moving average and pure autoregressive models have covariance matrices which are expressed in terms of two triangular matrices. Furthermore, the generalised variance is obtained from a factorisation of the determinant of the covariance matrix into four constituent parts. Examples of these theorems are given. The results are generalised for estimators of a mixed autoregressive moving average model in which there is either just one moving average parameter or just one autoregressive parameter. In particular the generalised variance is factorised into the determinants of the covariance matrices for efficient estimators of the parameters of the corresponding two pure models, and two other scalar terms. The submatrices of the covariance matrix for the efficient estimators of the parameters of the general mixed model are found by specifying four or five upper triangular matrices, whose non-zero elements are single parameters of the model, and then carrying out some matrix multiplications and additions. Provided the model is not too large, explicit expressions for the variances and covariances can be obtained. Examples, using mixed models, of these methods are given, and the adequacy of the fitted model is discussed in detail. It is proposed that these theorems enable statistical tests to be applied to problems of active tracking, which, traditionally, are expressed in terms of polynomial-projecting dynamic linear models. The problem of testing for constant velocity is considered in detail. A test based on a generalisation of Student's t test is discussed. Several examples of this test procedure are given.
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in Royal Holloway Research Online.Last modified on 01-Feb-2017
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Institution: University of London, Royal Holloway College (United Kingdom).