Solved – Does the UMVUE have to be a minimal sufficient statistic

I'm studying point estimation and I have found this question that seems pretty tricky to me.

If $T$ is a minimal sufficient statistic for $theta$ with $E(T) = tau(theta)$, can you say that $T$ is also the UMVUE for $tau(theta)$?

Rao-Blackwell theorem states that an unbiased estimator $T$ for $tau(theta)$ can be improved using a sufficient statistic $U$ for $theta$, i.e. $T^*=E[T|U]$ has a variance lower than the one of $T$.

Lehmann-Scheffé theorem states that $T$ must be a function of a complete sufficient statistic in order to be the unique UMVUE for $tau(theta)$.

But what about the fact that $T$ is minimal sufficient? Does this provide some results about $T$?

Since a minimal sufficient statistic is not a complete statistic in general (cf. Is a minimal sufficient statistic also a complete statistic), the answer to your question is negative.

For a concrete example, consider $Xsim U(theta,theta+1)$ where $theta$ is the parameter of interest. Here $X$ is minimal sufficient for $theta$ but $X$ is not the UMVUE of its expectation. For details see this post.

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