# Optimal unbiased estimator

I have the sample $$X_1,...,X_n$$ of i.i.d. from $$U(\theta - 1/2; \theta +1/2)$$. It is well known that $$T = (X_{(1)}; X_{(n)})$$ is a sufficient but not complete statistic, because $$X_{(n)}-X_{(1)} - (n-1)/(n+1)$$ is unbiased estimator of zero.

I want to find unbiased estimator of $$\theta$$ with minimal variance. I have crude unbiased estimator of $$\theta$$ : $$\hat {\theta} = X_1.$$ From Blackwell-Rao-Kolmogorov theorem estimator $$\hat {\theta}_1 = E[X_1|X_{(1)} = t_1; X_{(n)} = t_2] = \frac{t_1 + t_2}{2}$$ is unbiased and uniformlly better than $$\hat {\theta} = X_1$$.

But because of completeless $$T = (X_{(1)}; X_{(n)})$$ this is not the only unbiased estimator as the function of $$T$$. Moreover if I iterate the Blackwell-Rao-Kolmogorov theorem with this new estimator $$\hat {\theta}_1 =\frac{t_1 + t_2}{2}$$ I don't improve it: $$E[\frac {X_{(1)} + X_{(n)}}{2}|X_{(1)} = t_1; X_{(n)} = t_2] = \frac{t_1 + t_2}{2} = \hat {\theta}_1.$$

Is the $$\frac {X_{(1)} + X_{(n)}}{2}$$ the optimal unbiased estimator of $$\theta$$ (but not unique)?