Ww know $T(x) = (X_{(1)}, X_{(n)})$ for a random sample from the uniform distribution on $(\theta, 2\theta)$ with $\theta \gt 0$ is minimal sufficient for $\theta$. Show why sample mean and sample variance $(\bar X, S^2)$ are not sufficient for $\theta$
Any hint or help is welcome
We know that $(X_{(1)},X_{(n)})$ is minimal sufficient for $\theta$. Recall that $T(X)$ is minimal sufficient iff, for any sufficient $S$, then $T=\psi(S)$ a.s. in the whole family $\mathcal{U}(\theta,2\theta)$ for a Borel $\psi$. Now suppose that $(\overline{X},S^2)$ is sufficient; then there should be a $\psi$ s.t. $(X_{(1)},X_{(n)})=\psi(\overline{X},S^2)$; however there is no such $\psi$, which would imply that $(X_{(1)},X_{(n)})$ is not minimal sufficient (but it is). We conclude that $(\overline{X},S^2)$ is not sufficient.