A Least Squares Solution of Linear Equations with Coefficients Subject to a Special Type of Error

Abstract

Following the classical method a least squares solution is given for the equations $$ where the ark and brk are fixed known constants and the er are observed values subject to error. The solution is obtained as a series in the successive moments of the joint distribution of the er, and only terms up to those involving the variance are retained. In this approximation the estimated values of the xk are biased, but, after correction for this bias and using a particular weight for each equation, the classical tests of significance for the case brk = 0 can be applied unchanged. With suitable assumptions it is shown that the series converges more and more rapidly as n ? 8 for almost all sequences of the er