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That is, if $T$ is UMVUE for $\theta$, does this mean that $T$ is a complete and sufficient statistic for $\theta$?

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    $\begingroup$ I believe the sample median of a sample drawn from the Laplace distribution provides a counter example but it needs to take me more time to write down a rigorous proof. I will come back when I got time. $\endgroup$
    – Zhanxiong
    Commented Aug 16, 2017 at 5:28

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