Let us consider a simple model.
$y_i = \beta + \epsilon_i$
If we assume that $\epsilon_i$ has 0 mean, constant variance and is uncorrelated. Then via Gauss-Markov theorem we know that $\hat{\beta} = \bar{y}$ is the BLUE.
However, when I assume that $\epsilon_i \sim^{iid} U(-\sigma,\sigma)$ then I am getting $\hat{\beta}_{UMVUE}=\frac{y_{(1)}+y_{(n)}}{2}$.
Isn't $\hat{\beta}_{UMVUE}$ Linear? If so isn't it a violation of the Gauss-Markov Theorem? Where am I going wrong?
Also can anyone suggest me a distribution of $\epsilon$ where I can get a Non-Linear unbiased estimator (in closed form) which is better than OLS.