8The Gauss‐Markov Theorem
8.1 A Simple Example
The Gauss‐Markov theorem is named by tradition after two mathematicians, one German and one Russian,1who were however not collaborators and whose lives did not even overlap. It is the famous result that the least squares estimator is efficient in the class of linear unbiased estimators in the regression model.
The efficiency of an estimator is the property that its variance with respect to the sampling distribution is the smallest in the specified class. The comparison is between alternative estimators based on a given sample. Thus, it is incorrect usage to say that least squares in a large sample is more efficient than in a small sample, even though the variance may be smaller in the former case. The task is to compare the sampling distributions of different estimators of 
in (7.1) for a given design matrix 
.
These ideas will be developed first in the simplest possible example, the mean 
of a random sample 
. If 
and for each and Cov for , assumptions ...
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