= Support-restricted confidence interval for a uniform location parameter
{title2=$[U_{(2)}-a,\ U_{(1)}+a]$}
For two independent observations uniform on $[\theta-a,\theta+a]$, the displayed interval of possible parameter values contains the true parameter almost surely. Intersecting any <confidence interval> with it preserves that interval's coverage, while excluding values inconsistent with the observations. The <order statistics> interval $[U_{(1)},U_{(2)}]$ has coverage $1/2$, since the observations straddle $\theta$ with probability $1/2$. If their separation exceeds $a$, the entire interval of possible parameters lies between the observations; this explains why the nominal one-half procedure can be unnecessarily wide on that event.
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