BIB-VERSION:: CS-TR-v2.0
          ID:: STAN//CS-TR-71-236
       ENTRY:: November 01, 1995
ORGANIZATION:: Stanford University, Department of Computer Science
       TITLE:: Numerical computations for univariate linear models
        TYPE:: Technical Report
      AUTHOR:: Golub, Gene H.
      AUTHOR:: Styan, George P. H.
        DATE:: September 1971
       PAGES:: 39
    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 $\underset ~\to L
               '\underset ~\to \gamma\ = \underset ~\to 0$.
       NOTES:: [Adminitrivia V1/Prg/19951101]
         END:: STAN//CS-TR-71-236