Numerical computations for univariate linear models

Abstract

We consider the usual univariate linear model In Part One of this paper X has full column rank. Numerically stable and efficient computational procedures are developed for the least squares estimation of y and the error sum of squares. We employ an orthogonal triangular decomposition of X using Householder Transformations. A lower bomd for the condition number of X is immediately obtained from this decomposition. Similar computational procedures are presented for the usual F-test of the general linear hypothesis L′γ=0; L′γ=m is also considered for m≠0. Updating techniques are given for adding to or removing from (X,y) a row, a set of rows or a column. In Part Two, X has less than full rank. Least squares estimates are obtained using generalized inverses. The function L′γ is estimable whenever it admits an unbiased estimator linear in y. We show how to computationally verify esthabiiity of L′γ and the equivalent testability of L′γ=0.