1. 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. The Gauss{Markov Theorem. On best linear estimation and general Gauss--Markov theorem in linear models with arbitrary nonnegative covariance structure. (An Alternative Statement… The assumptions under which this statement is true include all but normality; i.e., the statement is still true when The so-called Gauss-Markov theorem states that under certain conditions, least-squares estimators are “best linear unbiased estimators” (“BLUE”), “best” meaning having minimum variance in the class of unbiased linear estimators. The Gauss-Markov theorem states that the OLS estimator is the most efficient. The theorem states that out of the class of estimators that are linear in Y, OLS is the “Best” where “Best” refers to the smallest variance of the estimated coefficients. To prove this, take an arbitrary linear, unbiased estimator $\bar{\beta}$ of $\beta$. It is rather surprising that the second algebraic result is usually derived in a differential way. It is worthwhile to consider an alternative statement and proof of the theorem, which also considers the variance of an arbitrary linear combination of the elements of ﬂ^. In statistics, the Gauss–Markov theorem, named after Carl Friedrich Gauss and Andrey Markov, states that in a linear regression model in which the errors have expectation zero and are uncorrelated and have equal variances, the best linear unbiased estimator (BLUE) of the coefficients is given by the ordinary least squares (OLS) estimator. Then criticize it. ↑ Department of Mathematics and Statistics, FI-33014 University of Tampere, Tampere, Finland. Wikipedia’s stated pre-conditions of the Gauss-Markov theorem. The Gauss-Markov theorem is a very strong statement. The concept of … In statistics, the Gauss–Markov theorem, named after Carl Friedrich Gauss and Andrey Markov, states that in a linear regression model in which the errors have expectation zero, are uncorrelated and have equal variances, the best linear unbiased estimator (BLUE) of the coefficients is given by the ordinary least squares (OLS) estimator, provided it exists. Earlier, one of the desirable properties of estimators was that the estimator has minimum variance. The Gauss-Markov theorem states that, under the usual assumptions, the OLS estimator $\beta_{OLS}$ is BLUE (Best Linear Unbiased Estimator). Gauss-Markov theorem reduces linear unbiased estimation to the Least Squares Solution of inconsistent linear equations while the normal equations reduce the second one to the usual solution of consistent linear equations. Why do we care about linearity? To apply the Gauss-Markov theorem the Wikipedia says you must assume your data has the following properties: E[e(i)] = 0 (lack of structural errors, needed to avoid bias) The Gauss-Markov Theorem for the transformed model implies that the BLUE of b for the generalized regression model is the OLS estimator applied to (6): bˆ GLS = (X 0X) 1X0y = (X0P0PX) 1X0P0Py = (X0W 1X) 1X0W 1y (7) This is the GLS estimator. of the Gauss{Markov Theorem that uses this measure can also be proved. We are going to read through the Wikipedia statement of the Gauss-Markov theorem in detail. SIAM Journal on Applied Mathematics , 17 , 1190--1202. Solution: The G-M theorem states that, among all linear unbiased estimators of the regression parameters, the ordinary least squares estimates have minimum variance. (15) State the Gauss-Markov theorem. 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