4.II.12H $\quad$

State and prove the Rao-Blackwell theorem.

Suppose that $X_{1}, \ldots, X_{n}$ are independent random variables uniformly distributed over $(\theta, 3 \theta)$. Find a two-dimensional sufficient statistic $T(X)$ for $\theta$. Show that an unbiased estimator of $\theta$ is $\hat{\theta}=X_{1} / 2$.

Find an unbiased estimator of $\theta$ which is a function of $T(X)$ and whose mean square error is no more than that of $\hat{\theta}$.

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