# Trouble with Gauss-Markov Theorem and with finding a Best “Non-Linear” Unbiased Estimator

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.

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.