Report Number: CS-TR-71-236
Institution: Stanford University, Department of Computer Science
Title: Numerical computations for univariate linear models
Author: Golub, Gene H.
Author: Styan, George P. H.
Date: September 1971
Abstract: We consider the usual univariate linear model E($\underset ~\to y$) = $\underset ~\to X \underset ~\to \gamma$ , V ($\underset ~\to y$) = $\sigma^2 \underset ~\to I$. In Part One of this paper $\underset ~\to X$ has full column rank. Numerically stable and efficient computational procedures are developed for the least squares estimation of $\underset ~\to \gamma$ and the error sum of squares. We employ an orthogonal triangular decomposition of $\underset ~\to X$ using Householder transformations. A lower bound for the condition number of $\underset ~\to X$ is immediately obtained from this decomposition. Similar computational procedures are presented for the usual F-test of the general linear hypothesis $\underset ~\to L\ ' \underset ~\to \gamma$ = $\underset ~\to 0$ ; $\underset ~\to L\ ' \underset ~\to \gamma$ = $\underset ~\to m$ is also considered for $\underset ~\to m\ \neq\ 0$. Updating techniques are given for adding to or removing from ($\underset ~\to X ,\underset ~\to y$) a row, a set of rows or a column . In Part Two, $\underset ~\to X$ has less than full rank. Least squares estimates are obtained using generalized inverses. The function $\underset ~\to L '\underset ~\to \gamma$ is estimable whenever it admits an unbiased estimator linear in $\underset ~\to y$. We show how to computationally verify estimability of $\underset ~\to L '\underset ~\to \gamma$ and the equivalent testability of $\underset ~\to L '\underset ~\to \gamma\ = \underset ~\to 0$.