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 $\underset ~\to L
'\underset ~\to \gamma\ = \underset ~\to 0$.