# The relationship between UMVUE and complete sufficient statistic

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?

• You might refer to the last example in this note. – StubbornAtom Jun 30 '18 at 20:19
• I have asked a similar question:stats.stackexchange.com/questions/353431/…. – StubbornAtom Jun 30 '18 at 20:21
• @StubbornAtom Hi . I think here we need to consider the relationship between UMVUE and unbiased estimator of zero. – student_R123 Jul 19 '18 at 19:58
• Hi, did you actually verify that a complete sufficient statistic does not exist? By my calculations, $\max_{1\le i\le n}|X_i|$ is a complete sufficient statistic for $\theta$. If so, UMVUE of $\theta$ would naturally exist. – StubbornAtom Jul 24 '18 at 16:23
• Yes, it is .... – StubbornAtom Aug 4 '18 at 14:27