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.