Gauss-Markov theorem
This article is 'not about Gauss-Markov processes.
In statistics, the 'Gauss-Markov theorem' states that in a linear model in which the errors have expectation zero and are uncorrelated and have equal variances, the best linear unbiased estimators of the coefficients are the least-squares estimators. More generally, the best linear unbiased estimator of any linear combination of the coefficients is its least-squares estimator. The errors are not assumed to be normally distributed, nor are they assumed to be independent (but only uncorrelated --- a weaker condition), nor are they assumed to be identically distributed (but only homoscedastic --- a weaker condition, defined below).
More explicitly, and more concretely, suppose we have
for i = 1, . . . , n, where β0 and β1 are non-random but unobservable parameters, xi are non-random and observable, εi are random, and so Yi are random. (We set x in lower-case because it is not random, and Y in capital because it is random.) The random variables εi are called the "errors". The Gauss-Markov assumptions state that
(i.e., all errors have the same variance; that is "homoscedasticity"), and
for  ; that is "uncorrelatedness."
A linear unbiased estimator of β1 is a linear combination
in which the coefficients ci are not allowed depend on the earlier coefficients βi, since those are not observable, but are allowed to depend on xi, since those are observable, and whose expected value remains β1 even if the values of βi change. (The dependence of the coefficients on the xi is typically nonlinear; the estimator is linear in that which is random; that is why this is "linear" regression.) The mean squared error of such an estimator is
i.e., it is the expectation of the square of the difference between the estimator and the parameter to be estimated. (The mean squared error of an estimator coincides with the estimator's variance if the estimator is unbiased; for biased estimators the mean squared error is the sum of the variance and the square of the bias.) The best linear unbiased estimator is the one with the smallest mean squared error. The "least-squares estimators" of β0 and β1 are the functions  and  of the Ys and the xs that make the sum of squares of residuals
as small as possible. (It is easy to confuse the concept of error introduced early in this article, with this concept of residual. For an account of the differences and the relationship between them, see errors and residuals in statistics.)
The main idea of the proof is that the least-squares estimators are uncorrelated with every linear unbiased estimator of zero, i.e., with every linear combination
whose coefficients do not depend upon the unobservable βi but
whose expected value remains zero regardless of how the values of β1 and β2 change.
External links
For a brief history of the theorem and an explanation of its name see the entry on the Gauss-Markov theorem in
Referenced By
C. F. Gauss | Carl F. Gauss | Carl Frederich Gauss | Carl Friedrich Gauss | Carl Gauss | Estimator | Gauss-Markov | Gauss-Markov process | Johann Carl Friedrich Gauss | Karl Friedrich Gauss | Karl Gauss | Least-squares method | Least squares | Linear Regression | List of mathematical proofs | List of mathematical topics (G-I) | List of mathematical topics (G-Z) | List of probability topics | List of proofs | List of statistical topics | Method of least squares | ProbabilityApplications | Probability Applications
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