Prove that the Pitman estimator is itself complete sufficient

It's a homework question, but I just have no idea about it ...

Let $X_{1} ,... , X_{n}$ be random variables according to a distribution having joint density $f(x_{1}-\theta ,...,x_{n}-\theta )$,where $\theta \in R$ is a location parameter. Assume that there exists a complete sufficient statistics $S(X_{1} ,... , X_{n} )$ for $\theta$. Prove that the Pitman estimator

$\delta^{*}(X_{1},...,X_{n})=\frac{\int _{R}\theta f(X_{1}-\theta ,...,X_{n}-\theta) d\theta }{\int _{R} f(X_{1}-\theta ,...,X_{n}-\theta) d\theta }$

is itself complete sufficient.

• Add the self-study tag. – Michael Chernick Jun 5 at 3:20
• Please use MathJax for typesetting math here. – StubbornAtom Jun 6 at 13:10