GaussMarkov theorem
A GaussMarkov linear model is a linear statistical model that satisfies all the conditions of a general linear model except the normality of the error terms. Formally, if $\bm{Y}$ is an $m$dimensional response variable vector, and ${\bm{Z}}_{\bm{i}}={z}_{i}(\bm{X})$, $i=1,\mathrm{\dots},k$ are the $m$dimensional functions of the explanatory variable vector $\bm{X}$, a GaussMarkov linear model has the form:
$$\bm{Y}={\beta}_{0}{\bm{Z}}_{\mathrm{\U0001d7ce}}+\mathrm{\cdots}+{\beta}_{k}{\bm{Z}}_{\bm{k}}+\mathit{\u03f5},$$ 
with $\mathit{\u03f5}$ the error vector such that

1.
$\mathrm{E}[\mathit{\u03f5}]=\mathrm{\U0001d7ce}$, and

2.
$\mathrm{Var}[\mathit{\u03f5}]={\sigma}^{2}\bm{I}$.
In other words, the observed responses ${Y}_{i}$, $i=1,\mathrm{\dots},m$ are not assumed to be normally distributed, are not correlated with one another, and have a common variance^{} $\mathrm{Var}[{Y}_{i}]={\sigma}^{2}$.
GaussMarkov Theorem. Suppose the response variable $\bm{Y}=({Y}_{1},\mathrm{\dots},{Y}_{m})$ and the explanatory variables $\bm{X}$ satisfy a GaussMarkov linear model as described above. Consider any linear combination^{} of the responses
$Y={\displaystyle \sum _{i=1}^{m}}{c}_{i}{Y}_{i},$  (1) 
where ${c}_{i}\in \mathbb{R}$. If each ${\mu}_{i}$ is an estimator^{} for response ${Y}_{i}$, parameter $\theta $ of the form
$\theta ={\displaystyle \sum _{i=1}^{m}}{c}_{i}{\mu}_{i},$  (2) 
can be used as an estimator for $Y$. Then, among all unbiased estimators^{} for $Y$ having form (2), the ordinary least square estimator (OLS)
$\widehat{\theta}={\displaystyle \sum _{i=1}^{m}}{c}_{i}\widehat{{\mu}_{i}},$  (3) 
yields the smallest variance. In other words, the OLS estimator is the uniformly minimum variance unbiased estimator.
Remark. $\widehat{\theta}$ in equation (3) above is more popularly known as the BLUE, or the best linear unbiased estimator for a linear combination of the responses in a GaussMarkov linear model.
Title  GaussMarkov theorem 

Canonical name  GaussMarkovTheorem 
Date of creation  20130322 15:02:53 
Last modified on  20130322 15:02:53 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  8 
Author  CWoo (3771) 
Entry type  Theorem 
Classification  msc 62J05 
Synonym  BLUE 
Related topic  LinearLeastSquaresFit 
Defines  GaussMarkov linear model 
Defines  best linear unbiased estimator 