A single scalar $\beta$ is measured by $m$ different people. Assuming uncorrelated measurement errors of equal variance, show that the minimum-variance unbiased estimate of $\beta$ is equal to the average of the $m$ measurement values.
Solution:
I tried to use Gauss-Markov theorem but I have no guarantee that the measures are uncorrelated as well.
Source: Optimization by Vector Space Methods. Luenberger. Ch4. Prob 1.
The model is given by:
$$ {y}_{i} = \beta + {e}_{i} $$
Where $ {y}_{i} $ is the $ i $ -th measurement and $ {e}_{i} $ is the error of the $ i $ -th measurement.
In matrix form it becomes:
$$ y = \beta \boldsymbol{1} + e $$
Now, this is the same model as in Page 84 of the book where $ W = \boldsymbol{1} $.
The estimator is given by $ \hat{\beta} = {k}^{T}y $ where $ {k}^{T} $ is given by:
$$ {k}^{T} = \left( \boldsymbol{1}^{T} {Q}^{-1} \boldsymbol{1} \right)^{-1} \boldsymbol{1}^{T} {Q}^{-1} $$
Since $ Q = {\sigma}_{n}^{2} I $ since noise is uncorrelated and with equal variance then the above:
$$ {k}^{T} = \left( \boldsymbol{1}^{T} {Q}^{-1} \boldsymbol{1} \right)^{-1} \boldsymbol{1}^{T} {Q}^{-1} = \frac{1}{n} \boldsymbol{1}^{T} $$
This is expexted as we expect the answer to be the average of samples.