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Gauss–Markov theorem
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<p>In <a href="/facts/Statistics/DNPsKGYU">statistics</a>, the Gauss–Markov theorem (or simply Gauss theorem for some authors) states that the <a href="/facts/Ordinary_least_squares/q7H9k5vM">ordinary least squares</a> (OLS) estimator has the lowest <a href="/facts/Sampling_variance/3k8Fjlb3">sampling variance</a> within the <a href="/facts/Class_(set_theory)/F5EmDd9b">class</a> of <a href="/facts/Linear_combination/CVjL6kuN">linear</a> <a href="/facts/Bias_of_an_estimator/oxIvEgmd">unbiased</a> <a href="/facts/Estimator/CbkjcKvN">estimators</a>, if the <a href="/facts/Errors_and_residuals/Ef0LgAS4">errors</a> in the <a href="/facts/Linear_regression_model/5n998IhK">linear regression model</a> are <a href="/facts/Uncorrelated/Ub2XK1WJ">uncorrelated</a>, have <a href="/facts/Homoscedasticity/50v2qyEN">equal variances</a> and expectation value of zero. The errors do not need to be <a href="/facts/Normal_distribution/UapjjPyQ">normal</a>, nor do they need to be <a href="/facts/Independent_and_identically_distributed/othIRaWt">independent and identically distributed</a> (only <a href="/facts/Uncorrelated/Ub2XK1WJ">uncorrelated</a> with mean zero and <a href="/facts/Homoscedastic/50v2qyEN">homoscedastic</a> with finite variance). The requirement that the estimator be unbiased cannot be dropped, since biased estimators exist with lower variance. See, for example, the <a href="/facts/James%25E2%2580%2593Stein_estimator/o7tcLXmt">James–Stein estimator</a> (which also drops linearity), <a href="/facts/Ridge_regression/Le6KJgJ1">ridge regression</a>, or simply any <a href="/facts/Degenerate_distribution/Tj5b5h2z">degenerate</a> estimator. </p><p>The theorem was named after <a href="/facts/Carl_Friedrich_Gauss/f4eSMBD7">Carl Friedrich Gauss</a> and <a href="/facts/Andrey_Markov/F409PDqD">Andrey Markov</a>, although Gauss' work significantly predates Markov's. But while Gauss derived the result under the assumption of independence and normality, Markov reduced the assumptions to the form stated above. A further generalization to <a href="/facts/Heteroscedasticity/msBQWI7G">non-spherical errors</a> was given by <a href="/facts/Alexander_Aitken/UbhEranD">Alexander Aitken</a>. </p>