A not complete Statistics

Question

Let $X_1, \dots,X_n$ be i.i.d. from the uniform distribution on the interval $(0, \theta)$. How can I show that for $\theta>1$ the $X_{(n)}$ statistic is not complete?

I know and that if $\theta>0$ the $X_{(n)}$ statistic is complete. But I don't know how to show the case in $\theta>1$. I tried to show this using $E[g(X_{(n)}]=0$, but it is not working.

Any help?

Answer 1

Let $g(t)=\frac{\cos(\pi t)}{nt^{n-1}}1_{(0,1)}(t)$, then $\mathsf{E}_\theta[g(X_{(n)})]=0$, $\forall \theta>1$.