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{{Redirect|BLUE|queue management algorithm|Blue (queue management algorithm)}}
{{Regression bar}}
 
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 [[variance]]s, the '''best linear [[bias of an estimator|unbiased]] [[estimator]]''' ('''BLUE''') of the coefficients is given by the [[ordinary least squares]] (OLS) estimator. Here "best" means giving the lowest variance of the estimate, as compared to other unbiased, linear estimates. The errors don't need to be [[normal distribution|normal]], nor do they need to be [[independent and identically distributed]] (only [[uncorrelated]] and [[homoscedastic]]). The hypothesis that the estimator be unbiased cannot be dropped, since otherwise estimators better than OLS exist. See for examples the [[James–Stein estimator]] (which also drops linearity) or [[ridge regression]].
 
== Statement ==
Suppose we have in matrix notation,
:<math> \underline{y} = X \underline{\beta} + \underline{\varepsilon},\quad (\underline{y},\underline{\varepsilon} \in \mathbb{R}^n, \beta \in \mathbb{R}^K \text{ and } X\in\mathbb{R}^{n\times K}) </math>
expanding to,
:<math> y_i=\sum_{j=1}^{K}\beta_j X_{ij}+\varepsilon_i \quad \forall i=1,2,\ldots,n</math>
 
where <math>\beta_j</math> are non-random but '''un'''observable parameters, <math> X_{ij} </math> are non-random and observable (called the "explanatory variables"), <math>\varepsilon_i</math> are  random, and so <math>y_i</math> are random. The random variables <math>\varepsilon_i</math> are called the "residuals" or "noise" (will be contrasted with "errors" later in the article; see [[errors and residuals in statistics]]). Note that to include a constant in the model above, one can choose to introduce the constant as a variable <math>\beta_{K+1}</math>  with a newly introduced last column of X being unity i.e., <math>X_{i(K+1)} = 1</math> for all <math> i </math>.
 
The '''Gauss–Markov''' assumptions are
 
*<math>E(\varepsilon_i)=0, </math>
*<math>V(\varepsilon_i)= \sigma^2 < \infty,</math>
(i.e., all residuals have the same variance; that is "[[homoscedasticity]]"), and
*<math>{\rm cov}(\varepsilon_i,\varepsilon_j) = 0, \forall i \neq j </math>
 
for <math> i\neq j</math>  that is, any the noise terms are drawn from an "uncorrelated" distribution. A '''linear estimator''' of <math> \beta_j  </math> is a linear combination
 
:<math>\widehat\beta_j = c_{1j}y_1+\cdots+c_{nj}y_n</math>
 
in which the coefficients <math> c_{ij} </math>  are not allowed to depend on the underlying coefficients <math> \beta_j </math>, since those are not observable, but are allowed to depend on the values <math> X_{ij} </math>, since these data are observable. (The dependence of the coefficients on each <math> X_{ij} </math> is typically nonlinear; the estimator is linear in each <math> y_i </math>  and hence in each random <math> \varepsilon </math>, which is why this is [[linear regression|"linear" regression]].)  The estimator is said to be '''unbiased''' [[if and only if]]
 
:<math>E(\widehat\beta_j)=\beta_j\,</math>
 
regardless of the values of <math> X_{ij} </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
 
:<math>E \left(\left(\sum_{j=1}^K\lambda_j(\widehat\beta_j-\beta_j)\right)^2\right);</math>
 
i.e., it is the expectation of the square of the weighted sum (across parameters) of the differences between the estimators and the corresponding parameters to be estimated. (Since we are considering the case in which all the parameter estimates are unbiased, this mean squared error is the same as the variance of the linear combination.) The '''best linear unbiased estimator''' (BLUE) of the vector <math> \beta </math>  of parameters <math> \beta_j </math>  is one with the smallest mean squared error for every vector <math> \lambda </math> of linear combination parameters.  This is equivalent to the condition that
 
:<math>V(\tilde\beta)- V(\widehat\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'X)^{-1}X'y</math>
 
of <math> y </math> and <math> X </math> (where <math>X'</math> denotes the [[transpose]] of <math> X </math>)
that minimizes the '''sum of squares of [[errors and residuals in statistics|residuals]]''' (misprediction amounts):
 
:<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>
 
The theorem now states that the OLS estimator is a BLUE. The main idea of the proof is that the least-squares estimator is uncorrelated 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 <math> \beta </math> but whose expected value is always zero.
 
== Proof ==
Let <math> \tilde\beta = Cy </math> be another linear estimator of <math> \beta </math> and let ''C'' be given by <math> (X'X)^{-1}X' + D </math>, where ''D'' is a <math>k \times n</math> nonzero matrix. As we're restricting to ''unbiased'' estimators, minimum mean squared error implies minimum variance. The goal is therefore to show that such an estimator has a variance no smaller than that of <math> \hat\beta </math>, the OLS estimator.
 
The expectation of <math> \tilde\beta </math> is:
:<math>
\begin{align}
E(Cy) &= E(((X'X)^{-1}X' + D)(X\beta + \varepsilon)) \\
&= ((X'X)^{-1}X' + D)X\beta + ((X'X)^{-1}X' + D)\underbrace{E(\varepsilon)}_0 \\
&= (X'X)^{-1}X'X\beta + DX\beta \\
&= (I_k + DX)\beta. \\
\end{align}
</math>
 
Therefore, <math> \tilde\beta </math> is unbiased if and only if <math> DX = 0 </math>.
 
The variance of <math> \tilde\beta </math> is
:<math>
\begin{align}
V(\tilde\beta) &= V(Cy) = CV(y)C' = \sigma^2 CC' \\
&= \sigma^2((X'X)^{-1}X' + D)(X(X'X)^{-1} + D') \\
&= \sigma^2((X'X)^{-1}X'X(X'X)^{-1} + (X'X)^{-1}X'D' + DX(X'X)^{-1} + DD') \\
&= \sigma^2(X'X)^{-1} + \sigma^2(X'X)^{-1} (\underbrace{DX}_{0})' + \sigma^2 \underbrace{DX}_{0} (X'X)^{-1} + \sigma^2DD' \\
&= \underbrace{\sigma^2(X'X)^{-1}}_{V(\hat\beta)} + \sigma^2DD'.
\end{align}
</math>
 
Since ''DD''' is a positive semidefinite matrix, <math> V(\tilde\beta) </math> exceeds <math> V(\hat\beta) </math> by a positive semidefinite matrix.
 
== Generalized least squares estimator ==
The [[generalized least squares]] (GLS) or [[Alexander Aitken|Aitken]] estimator extends the Gauss–Markov theorem to the case where the error vector has a non-scalar covariance matrix{{spaced ndash}}the Aitken estimator is also a BLUE.<ref>A. C. Aitken, "On Least Squares and Linear Combinations of Observations", ''Proceedings of the Royal Society of Edinburgh'', 1935, vol. 55, pp. 42–48.</ref>
 
==See also==
*[[Independent and identically distributed random variables]]
*[[Linear regression]]
*[[Measurement uncertainty]]
 
=== Other unbiased statistics ===
*[[Best linear unbiased prediction]] (BLUP)
*[[Minimum-variance unbiased estimator]] (MVUE)
 
==Notes==
<references />
 
==References==
{{refbegin}}
* {{cite journal
|authorlink=R. L. Plackett |last=Plackett |first=R.L.
|year=1950
|title=Some Theorems in Least Squares
|journal=[[Biometrika]]
|volume=37 |issue=1–2 |pages=149–157
|doi=10.1093/biomet/37.1-2.149  |mr=36980 | jstor = 2332158
}}
{{refend}}
 
==External links==
*[http://jeff560.tripod.com/g.html Earliest Known Uses of Some of the Words of Mathematics: G] (brief history and explanation of the name)
*[http://www.xycoon.com/ols1.htm Proof of the Gauss Markov theorem for multiple linear regression] (makes use of matrix algebra)
*[http://emlab.berkeley.edu/GMTheorem/index.html A Proof of the Gauss Markov theorem using geometry]
 
{{Least squares and regression analysis|state=expanded}}
 
{{DEFAULTSORT:Gauss-Markov theorem}}
[[Category:Statistical theorems]]

Latest revision as of 06:49, 22 August 2014

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