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# Gauss-Markov theorem

A *Gauss-Markov 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 $\boldsymbol{Y}$ is
an $m$-dimensional response variable vector, and
$\boldsymbol{Z_{i}}=z_{i}(\boldsymbol{X})$, $i=1,\ldots,k$ are the
$m$-dimensional functions of the explanatory variable vector
$\boldsymbol{X}$, a Gauss-Markov linear model has the form:

$\boldsymbol{Y}=\beta_{0}\boldsymbol{Z_{0}}+\cdots+\beta_{k}\boldsymbol{Z_{k}}+% \boldsymbol{\epsilon},$ |

with $\boldsymbol{\epsilon}$ the error vector such that

1. $\operatorname{E}[\boldsymbol{\epsilon}]=\boldsymbol{0}$, and

2. $\operatorname{Var}[\boldsymbol{\epsilon}]=\sigma^{2}\boldsymbol{I}$.

In other words, the observed responses $Y_{i}$, $i=1,\ldots,m$ are not assumed to be normally distributed, are not correlated with one another, and have a common variance $\operatorname{Var}[Y_{i}]=\sigma^{2}$.

Gauss-Markov Theorem. Suppose the response variable $\boldsymbol{Y}=(Y_{1},\ldots,Y_{m})$ and the explanatory variables $\boldsymbol{X}$ satisfy a Gauss-Markov linear model as described above. Consider any linear combination of the responses

$\displaystyle Y=\sum_{{i=1}}^{m}c_{i}Y_{i},$ | (1) |

where $c_{i}\in\mathbb{R}$. If each $\mu_{i}$ is an estimator for response $Y_{i}$, parameter $\theta$ of the form

$\displaystyle\theta=\sum_{{i=1}}^{m}c_{i}\mu_{i},$ | (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)

$\displaystyle\hat{\theta}=\sum_{{i=1}}^{m}c_{i}\hat{\mu_{i}},$ | (3) |

yields the smallest variance. In other words, the OLS estimator is the uniformly minimum variance unbiased estimator.

Remark. $\hat{\theta}$ 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 Gauss-Markov linear
model.

## Mathematics Subject Classification

62J05*no label found*

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