Gauss–Markov theorem

WikiDoc Resources for Gauss–Markov theorem
Articles
Most recent articles on Gauss–Markov theorem Most cited articles on Gauss–Markov theorem Review articles on Gauss–Markov theorem Articles on Gauss–Markov theorem in N Eng J Med, Lancet, BMJ
Media
Powerpoint slides on Gauss–Markov theorem Images of Gauss–Markov theorem Photos of Gauss–Markov theorem Podcasts & MP3s on Gauss–Markov theorem Videos on Gauss–Markov theorem
Evidence Based Medicine
Cochrane Collaboration on Gauss–Markov theorem Bandolier on Gauss–Markov theorem TRIP on Gauss–Markov theorem
Clinical Trials
Ongoing Trials on Gauss–Markov theorem at Clinical Trials.gov Trial results on Gauss–Markov theorem Clinical Trials on Gauss–Markov theorem at Google
Guidelines / Policies / Govt
US National Guidelines Clearinghouse on Gauss–Markov theorem NICE Guidance on Gauss–Markov theorem NHS PRODIGY Guidance FDA on Gauss–Markov theorem CDC on Gauss–Markov theorem
Books
Books on Gauss–Markov theorem
News
Gauss–Markov theorem in the news Be alerted to news on Gauss–Markov theorem News trends on Gauss–Markov theorem
Commentary
Blogs on Gauss–Markov theorem
Definitions
Definitions of Gauss–Markov theorem
Patient Resources / Community
Patient resources on Gauss–Markov theorem Discussion groups on Gauss–Markov theorem Patient Handouts on Gauss–Markov theorem Directions to Hospitals Treating Gauss–Markov theorem Risk calculators and risk factors for Gauss–Markov theorem
Healthcare Provider Resources
Symptoms of Gauss–Markov theorem Causes & Risk Factors for Gauss–Markov theorem Diagnostic studies for Gauss–Markov theorem Treatment of Gauss–Markov theorem
Continuing Medical Education (CME)
CME Programs on Gauss–Markov theorem
International
Gauss–Markov theorem en Espanol Gauss–Markov theorem en Francais
Business
Gauss–Markov theorem in the Marketplace Patents on Gauss–Markov theorem
Experimental / Informatics
List of terms related to Gauss–Markov theorem

Please Take Over This Page and Apply to be Editor-In-Chief for this topic: There can be one or more than one Editor-In-Chief. You may also apply to be an Associate Editor-In-Chief of one of the subtopics below. Please mail us [1] to indicate your interest in serving either as an Editor-In-Chief of the entire topic or as an Associate Editor-In-Chief for a subtopic. Please be sure to attach your CV and or biographical sketch.

This article is not about Gauss–Markov processes.

Overview

In statistics, the Gauss–Markov theorem, named after Carl Friedrich Gauss and Andrey Markov, states that in a linear model in which the errors have expectation zero and are uncorrelated and have equal variances, a best linear unbiased estimator (BLUE) of the coefficients is given by the 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 having zero mean and equal variances).

Statement

Suppose we have

<math>Y_i=\sum_{j=1}^{K}\beta_j X_{ij}+\varepsilon_i</math>

for i = 1, . . ., n, where β_j are non-random but unobservable parameters, X_ij are non-random and observable (called the "explanatory variables"), ε_i are random , and so Y_i are random. The random variables ε_i are called the "errors" (not to be confused with "residuals"; see errors and residuals in statistics). Note that to include a constant in the model above, one can choose to include the X_iK = 1.

The Gauss–Markov assumptions state that

• <math>{\rm E}\left(\varepsilon_i\right)=0,</math>
• <math>{\rm Var}\left(\varepsilon_i\right)=\sigma^2<\infty,</math>

(i.e., all errors have the same variance; that is "homoscedasticity"), and

• <math>{\rm Cov}\left(\varepsilon_i,\varepsilon_j\right)=0</math>

for <math>i\not=j</math>; that is "uncorrelatedness." A linear estimator of β_j is a linear combination

<math>\widehat\beta_j = c_{1j}Y_1+\cdots+c_{nj}Y_n</math>

in which the coefficients c_ij are not allowed to depend on the earlier coefficients β, since those are not observable, but are allowed to depend on X, since this data is observable, and whose expected value remains β_j even if the values of X change. (The dependence of the coefficients on X is typically nonlinear; the estimator is linear in Y and hence in ε which is random; that is why this is "linear" regression.) The estimator is unbiased iff

<math>{\rm E}(\widehat\beta_j)=\beta_j.\,</math>

Now, let <math>\sum_{j=1}^K\lambda_j\beta_j</math> be some linear combination of the coefficients. Then the mean squared error of the corresponding estimation is defined as

<math>{\rm E} \left(\sum_{j=1}^K\lambda_j(\widehat\beta_j-\beta_j)^2\right)</math>

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.) A best linear unbiased estimator of β is the one with the smallest mean squared error for every linear combination λ. This is equivalent to the condition that

<math>{\rm Var}(\widehat\beta)-{\rm Var}(\tilde\beta)</math>

is a positive semi-definite matrix for every other linear unbiased estimator <math>\tilde\beta</math>.

The ordinary least squares estimator (OLS) is the function

<math>\widehat\beta=(X^{T}X)^{-1}X^{T}Y</math>

of Y and X that minimizes the sum of squares of residuals

<math>\sum_{i=1}^n\left(Y_i-\widehat{Y}_i\right)^2=\sum_{i=1}^n\left(Y_i-\sum_{j=1}^K\widehat\beta_j X_{ij}\right)^2.</math>

(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 theorem now states that the OLS estimator is a BLUE. The main idea of the proof is that the least-squares estimator isuncorrelated with every linear unbiased estimator of zero, i.e., with every linear combination <math>a_1Y_1+\cdots+a_nY_n</math> whose coefficients do not depend upon the unobservable β but whose expected value is always zero.

Generalized least squares estimator

The GLS or Aitken estimator extends the Gauss-Markov Theorem to the case where the error vector has a non-scalar covariance matrix - the Aitken estimator is also a BLUE.

See also

• Independent and identically-distributed random variables
• Linear regression
• Measurement uncertainty
• Best linear unbiased prediction

External links

• Earliest Known Uses of Some of the Words of Mathematics: G (brief history and explanation of its name)
• Proof of the Gauss Markov theorem for multiple linear regression (makes use of matrix algebra)
• A Proof of the Gauss Markov theorem using geometry

References

• Plackett, R.L. (1950) "Some Theorems in Least Squares", Biometrika 37: 149-157

Template:WS Template:Jb1