4.II.12H \quad

Statistics | Part IB, 2003

State and prove the Rao-Blackwell theorem.

Suppose that X1,,XnX_{1}, \ldots, X_{n} are independent random variables uniformly distributed over (θ,3θ)(\theta, 3 \theta). Find a two-dimensional sufficient statistic T(X)T(X) for θ\theta. Show that an unbiased estimator of θ\theta is θ^=X1/2\hat{\theta}=X_{1} / 2.

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

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