Master of Science (MS)
This work presents Gauss' justification of the method of least squares, following the treatment given by Gauss himself in "Theoria Combinationis Observationum Erroribus Minimis Obnoxiae," where the main idea is to show that the least squares estimate is the unbiased linear estimate of minimum variance. (Actually, we present Gauss' argument both in his terminology and translated into matrix terminology.) We show how this contrasts with Gauss' earlier justfication in "Theoria Motus Corporum Coelestium" which was based on the assumption of a normal distribution of errors, and yielded the estimate of maximum likelihood. We present as a background the development from scratch of all the probability theory needed, albeit we have not treated explicitly all the needed measure theory.
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Brand, Belinda B., "Gauss' method of least squares: an historically-based introduction" (2003). LSU Master's Theses. 2097.
James J. Madden