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COVARIANCES FOR FIXED INTERVAL SMOOTHED KALMAN FILTER PARAMETER ESTIMATES
Author(s) -
Haslett Stephen
Publication year - 1990
Publication title -
australian journal of statistics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.434
H-Index - 41
eISSN - 1467-842X
pISSN - 0004-9581
DOI - 10.1111/j.1467-842x.1990.tb01013.x
Subject(s) - mathematics , kalman filter , covariance , statistics , interval (graph theory) , smoothing , series (stratigraphy) , interval estimation , fixed point , gaussian , estimation theory , confidence interval , mathematical analysis , combinatorics , paleontology , physics , quantum mechanics , biology
Summary Given time series data for fixed interval t = 1,2,…, M with non‐autocorrelated innovations, the regression formulae for the best linear unbiased parameter estimates at each time t are given by the Kalman filter fixed interval smoothing equations. Formulae for the variance of such parameter estimates are well documented. However, formulae for covariance between these fixed interval best linear parameter estimates have previously been derived only for lag one. In this paper more general formulae for covariance between fixed interval best linear unbiased estimates at times t and t ‐ l are derived for t = 1,2,…, M and l = 0,1,…, t ‐ 1. Under Gaussian assumptions, these formulae are also those for the corresponding conditional covariances between the fixed interval best linear unbiased parameter estimates given the data to time M. They have application, for example, in determination via the expectation‐maximisation (EM) algorithm of exact maximum likelihood parameter estimates for ARMA processes expressed in statespace form when multiple observations are available at each time point.