Trouble with Gauss-Markov Theorem and with finding a Best "Non-Linear" Unbiased Estimator

Question

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.

Answer 1

I have found the answer to my own question. It's simple $\hat{\beta}_{UMVUE} = \frac{y_{(1)} + y_{(n)}}{2}$ is not linear. We can check that it doesn't satisfy the property of linear transformation that $T(x+y)=T(x)+T(y)$. Therefore Gauss-Markov theorem is not violated.In this case, i.e when $\epsilon \sim^{iid} U(-\sigma,\sigma)$, the estimator $\frac{y_{(1)} + y_{(n)}}{2}$ is the best unbiased estimator while $\bar{y}$ is the best linear unbiased estimator.