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The Rao-Blackwell theorem says that, for a sufficient statistic $T$ and a finite-second-moment estimator $\delta (X)$, the conditional estimator ${\delta }_T(T)=E_{\theta }[\delta (X)\mid T]$ can be chosen independently of the unknown parameter and has no greater mean squared error for estimating any specified function $g(\theta )$. Sufficiency makes the conditional distribution of the data given $T$ parameter-independent. Conditional expectation preserves the expectation, hence preserves unbiasedness. Writing $\delta -g=(\delta -{\delta }_T)+({\delta }_T-g)$, the cross term has expectation zero, because $E[\delta -{\delta }_T\mid T]=0$. ThereforeEquality holds exactly when $\delta ={\delta }_T$ almost surely. This proves the squared-loss form, including the usual unbiased-estimator variance statement.

Here $\theta >0$ is necessary for the stated uniform interval. Put $L=X_{(1)}={\min }_i{X}_i$ and $R=X_{(n)}={\max }_i{X}_i$. The joint density factors asThe factorization theorem for sufficient statistics shows that $T=(L,R)$ is sufficient. Also $E[X_1]=2\theta$, so $X_1/2$ is unbiased and has variance ${\theta }^2/12$.

For $n\ge 2$, conditional on distinct extrema $l,r$, exchangeability gives probability $1/n$ that $X_1$ is each endpoint; with probability $(n-2)/n$ it is one of the remaining observations, uniform on $(l,r)$. Its conditional mean is consequentlyThus the Rao-Blackwell estimator for a uniform scale interval iswhich is unbiased and has no greater mean squared error. For $n=1$, $L=R=X_1$ and the same formula reduces to the original estimator. As a quantitative check, the covariance of two uniform order statistics givesFor instance, after writing $X_i=\theta +2\theta U_i$, both extreme uniform variances are $n/[(n+1{)}^2(n+2)]$ and their covariance is $1/[(n+1{)}^2(n+2)]$. These formulas follow by integrating the joint extreme density $n(n-1)(r-l{)}^{n-2}$ on $0<l<r<1$; they also display strict improvement for $n>1$.