How can one proove, that $X$, which is uniformly distributed on the interval $(\theta, \theta+1), \theta \in \mathbb{R}$ is not complete for $\theta$?
I have to find a function, such that $E_{\theta}[h(T)] = 0$ where $h(t) \neq 0$ right? Does someone have an idea?
If I'm interpreting it right, then $X \sim \mathcal{U}(\theta,\theta+1)$ and the statistic we're considering is $T(X) = X$.
Note that $E[h(T(X)); \theta] =E[h(X);\theta]= \int_\theta^{\theta+1} h(x) \; dx$. Just need $h$ to integrate to zero over this region (and not be zero itself!). Think trigonometric functions . . .