I have done many problems with the Rao Blackwell Theory and finding an unbiased estimator, but this question has stumped me a little
Question
$Uniform(\theta, 3\theta)|\theta > 0$
$g_{\theta}(x) = \frac{1}{2\theta}$ $\theta\leq x\leq 3\theta$
Given $\hat{\theta}_{MOM} = \frac{1}{2}\bar{X}$ as unbiased for $\theta$
$T(\vec{X}) = (X_{1}, X_{n})$ is a sufficient statistic
Find using Rao-Blackwell, an unbiased estimator that is uniformly better than $\hat{\theta_{MOM}}$
But I am lost when we have a two-dimensional sufficient statistic instead of one.
Many thanks!