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)?


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