Updating linear models with dependent errors to include additional data and/or parameters

Abstract

Although updating equations for general linear models exist, they have been restricted to formulations with uncorrelated error structure, or to models involving fixed parameters of fixed length with correlated error structure. In this paper updating equations are developed for general linear models with correlated errors when the parameter vector is supplemented or evolves as further data are added. These equations form part of a general review in which updating formulae for minimum variance, linear, unbiased parameter estimates and their variance, applicable where a general linear model is altered by the inclusion of additional parameters and/or data, and where the variance matrices for the model error and the parameter vector are known and nonsingular, are outlined. When variance matrices are estimated rather than known, the equations derived give estimated generalized least squares estimators. The formulae are developed for both fixed and random parameter vectors. For models in which additional data and parameters are added simultaneously, it is shown that a generalization of the Kalman filter is possible that allows for arbitrary forms of model error autocorrelation, given known nonsingular error variance.