Let $X_1,...X_n$ $U(-\theta , \theta)$ I want to find the UMVUE of $\theta$ if it is exists.
My answer is , there is no UMVUE in this case.
Because there is no complete sufficient statistic that exists for $\theta$ . So although there exist an unbiased estimator of $\theta$, it is not a function of complete sufficient statistic.
So there is no there exist no UMVUE for $\theta$.
Am I correct in this situation?