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The theorem was named after Carl Friedrich Gauss and Andrey Markov, although Gauss' work significantly predates Markov's. [3] But while Gauss derived the result under the assumption of independence and normality, Markov reduced the assumptions to the form stated above. [4] A further generalization to non-spherical errors was given by Alexander ...
This transformation effectively standardizes the scale of and de-correlates the errors. When OLS is used on data with homoscedastic errors, the Gauss–Markov theorem applies, so the GLS estimate is the best linear unbiased estimator for .
The Gauss–Markov theorem shows that, when this is so, ^ is a best linear unbiased estimator . If, however, the measurements are uncorrelated but have different uncertainties, a modified approach might be adopted.
[a] [2] Ignoring simultaneity in the estimation leads to biased estimates as it violates the exogeneity assumption of the Gauss–Markov theorem. The problem of endogeneity is often ignored by researchers conducting non-experimental research and doing so precludes making policy recommendations. [3]
Gauss–Markov theorem; Mathematics portal; In mathematical optimization, the problem of non-negative least squares (NNLS) is a type of constrained least squares ...
An extended version of this result is known as the Gauss–Markov theorem. The idea of least-squares analysis was also independently formulated by the American Robert Adrain in 1808. In the next two centuries workers in the theory of errors and in statistics found many different ways of implementing least squares. [11]
Legendre and Gauss both applied the method to the problem of determining, from astronomical observations, the orbits of bodies about the Sun (mostly comets, but also later the then newly discovered minor planets). Gauss published a further development of the theory of least squares in 1821, [8] including a version of the Gauss–Markov theorem.
Gauss–Markov stochastic processes (named after Carl Friedrich Gauss and Andrey Markov) are stochastic processes that satisfy the requirements for both Gaussian processes and Markov processes. [1] [2] A stationary Gauss–Markov process is unique [citation needed] up to rescaling; such a process is also known as an Ornstein–Uhlenbeck process.