Show that if a function of a sufficient statistic is ancillary, then the sufficient statistic is not complete.

Question

I just proof that $T=(X_{(1)},X_{(n)})$ is not complete but now I want to show a more general case. To be more specific I want to show that if a function of a sufficient statistic is ancillary, then the sufficient statistic is not complete, because the expectation of that function doesn't depend on $\theta$ (the parameter). Any idea in how to proceed with this proof. I thought that since the Basu’s Theorem says that

"If $T(X)$ is a complete and minimal sufficient statistic, then $T(X)$ is independent of every ancillary statistic."

should be enough but I am not totally convinced of this.

Answer 1

Suppose that a function of a sufficient statistic $f(T)$ is ancillary. Then its pdf doesn't depend on $\theta$, call it $g_T$. $T$ is complete if for all $\theta$ $E_\theta(f(T))=c$ implies that $f(T)=c$. We have that $\int_a^b f(T)g_Td\theta=c$. This is $f(T)g_T(b-a)=c$, so $f(T)=\frac{c}{g_T(b-a)}$ which is not c. Therefore there exists a $\theta$ such that $E(f(T))=c$ but $f(T)=\frac{c}{g_T(b-a)}\ne c$. Hence T is not complete. Thus, if $f(T)$ is ancillary then $T$ is not complete.