  Practical or scientific considerations often lead to selecting a subset of
parameters as ``important.'' Inferences about those parameters often are based
on the same data used to select them in the first place. That can make the
reported uncertainties deceptively optimistic: confidence intervals that ignore
selection generally have less than their nominal coverage probability.
Controlling the probability that one or more intervals for selected parameters
do not cover---the ``simultaneous over the selected'' (SoS) error rate---is
crucial in many scientific problems. Intervals that control the SoS error rate
can be constructed in ways that take advantage of knowledge of the selection
rule. We construct SoS-controlling confidence intervals for parameters deemed
the most ``important'' $k$ of $m$ shift parameters because they are estimated
(by independent estimators) to be the largest. The new intervals improve
substantially over \v{S}id\'{a}k intervals when $k$ is small compared to $m$,
and approach the standard Bonferroni-corrected intervals when $k \approx m$.
Standard, unadjusted confidence intervals for location parameters have the
correct coverage probability for $k=1$, $m=2$ if, when the true parameters are
zero, the estimators are exchangeable and symmetric.
