Numerically stable computations for general univariate linear models

Abstract

We consider the general univariate linear model E(y) = Xb, V(y) = o² W, W symmetric nonnegative definite. A numerically stable method based on orthogonal rotations is given for computing the least squares estimate [bcirc] of b , as well as a representation of [/(b). It is shown how to extend the computations to update these results quickly and accurately when columns or rows of (y,X) are added or taken away. One of these techniques will handle the usual F-test for the general linear hypothesis, and the updating techniques can easily handle less than full rank X and W , while checking for consistency of the model. The first section describes some disadvantages of the original formulation of the problem and gives a general formulation which avoids these. The second section describes a numerically stable method for solution, while the third considers the statistical meaning of the computed quantities. Section 4 introduces the updating techniques as continuations of the original decomposition and Section 5 treats the special case of equality constraints

Keywords:

• regression analysis
• least squares
• uncorre-lated residuals
• covariance matrices
• equality constraints
• linear hypotheses
• updating techniques
• plane rotations