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\$?

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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.