4
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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?

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  • $\begingroup$ You might refer to the last example in this note. $\endgroup$ – StubbornAtom Jun 30 '18 at 20:19
  • $\begingroup$ I have asked a similar question:stats.stackexchange.com/questions/353431/…. $\endgroup$ – StubbornAtom Jun 30 '18 at 20:21
  • $\begingroup$ @StubbornAtom Hi . I think here we need to consider the relationship between UMVUE and unbiased estimator of zero. $\endgroup$ – student_R123 Jul 19 '18 at 19:58
  • $\begingroup$ 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. $\endgroup$ – StubbornAtom Jul 24 '18 at 16:23
  • 1
    $\begingroup$ Yes, it is .... $\endgroup$ – StubbornAtom Aug 4 '18 at 14:27

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