Assume $x_1x_2,\dots,x_n$ are i.i.d Uniform($\theta,\theta+1$), want to know if $x_{(n)}$, the order statistic, is a complete statistic.
This appears in a course slides, but I don't know if the result is right, since for $n=1$ the result is wrong. (by using the definition of complete statistic and the function $f(t)=sin(2\pi t)$).
any hint would be appreciated!