Suppose $X_1,...X_n$~$Unif([\theta,\theta+1])$ with unknown $\theta\in[0,\frac{1}{2}]$. I want to show that there does not exist an UMVU estimator for $g(\theta)=\theta+\frac{1}{2}$?
I first found two estimators $\hat{\gamma_1}=\frac{1}{2}+\frac{1}{n}\Sigma_{i=1}^{n}\mathbf{1}_{\{X_i\geqq1\}}$ and $\hat{\gamma_2}=\frac{1}{n}\Sigma_{i=1}^{n}\mathbf{1}_{\{X_i\geqq\frac{1}{2}\}}$, how can I show that for $\theta_0=0$ and $\theta_1=\frac{1}{2}$ I have $\mathbf{Var}_{\theta_j}[\hat{\gamma_j}]=0$? And how can I use this to show that no UMVU estimator exists for $g(\theta)$?