= Range-conditioned uniform location inference
For independent observations uniform on $(\theta-1/2,\theta+1/2)$, let $U$ and $V$ be the sample minimum and maximum, $R=V-U$, and $C=(U+V)/2$. For $n\ge2$ their joint <density> in $(c,r)$ is $n(n-1)r^{n-2}$ on $0<r<1$, $|c-\theta|<(1-r)/2$. Integrating over $c$ gives $f_R(r)=n(n-1)r^{n-2}(1-r)$, independent of $\theta$, so $R$ is an <ancillary statistic>. Moreover $C\mid R=r$ is uniform on $(\theta-(1-r)/2,\theta+(1-r)/2)$. Consequently
$$
\left[C-\frac{(1-\alpha)(1-R)}2,\ C+\frac{(1-\alpha)(1-R)}2\right]
$$
is an exact conditional $(1-\alpha)$ <confidence interval> for $\theta$. The realized range determines the appropriate precision, although the range is not itself evidence about the value of $\theta$.
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