for X~Uniform[a-1/2,a+1/2]

(X(1),X(n)) is minimal sufficient but it is not complete.

Does a complete statistic exist in this case?

There is a theorem that says that every complete sufficient statistic must be minimal sufficient.

Is it true to conclude that if there is a minimal sufficient statistic and it is not complete, then there is no complete statistic (since the complete statistic would be a minimal sufficient statistic and therefore a 1-1 function of that minimal sufficient statistic)?

Thanks

OK, I figured it out.

Found the following theorem

Theorem. If T is complete and sufficient for the family, and a minimal

sufficient statistic exists, then T is also minimal sufficient. (Lehmann &

Scheff´e 1950)

• If a minimal sufficient statistic is complete, then any minimal sufficient

statistic is complete.

•If a minimal sufficient statistic is not complete, then there are no

complete and sufficient statistics for the family.