Let $X_1, \cdot\cdot \cdot, X_n$ be a random sample from $U(\theta, \theta + 1)$, $\theta \in \mathbb{R}$.
Find a complete and minimal sufficient statistic $Y$ for $\theta$.
I proved that $T(X) = (X_{(1)}, X_{(n)})$ is a minimal sufficient statistics for $\theta$. However, it can be shown that $T(X)$ is not complete.
Any hints to find a complete sufficient statistics?
Many thanks!
You can find this problem in https://www-math.umd.edu/images/pdfs/quals/Statistics/Statistics-PhD-August-2022.pdf