Updates of statistics in a general linear model: a statistical interpretation and applications

We consider a general linear model (y,X,U) where U is a general positive definite matrix and X is possibly rank-deficient. We give updated formulae for various statistical quantities of interest (BLUEs, residual sum of squares, etc.) in the following situations: introduction of an additional observation, deletion of an observation, inclusion of a new regressor and deletion of a regressor. We give the formulae in statistical terminology so that their significance is better understood. We then give an application of these results to Regression Diagnostics in the linear model with correlated errors