= Rao-Blackwell estimator for a uniform scale interval
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{title2=$\widetilde\theta=(X_{(1)}+X_{(n)})/4$}
For independent <uniform random variables> on $(\theta,3\theta)$ with $\theta>0$, the sample minimum and maximum form a <sufficient statistic>. Conditional on these endpoints, <exchangeability> and the uniform law of the remaining observations give $E[X_1\mid X_{(1)},X_{(n)}]=(X_{(1)}+X_{(n)})/2$. Applying the <Rao-Blackwell theorem> to $X_1/2$ gives the displayed <unbiased estimator>. Its <variance> is $\theta^2/[2(n+1)(n+2)]$, compared with $\theta^2/12$ for $X_1/2$.
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