  In this paper we study the consistency of different bootstrap procedures for
constructing confidence intervals (CIs) for the unique jump discontinuity
(change-point) in an otherwise smooth regression function in a stochastic
design setting. This problem exhibits nonstandard asymptotics and we argue that
the standard bootstrap procedures in regression fail to provide valid
confidence intervals for the change-point. We propose a version of smoothed
bootstrap, illustrate its remarkable finite sample performance in our
simulation study, and prove the consistency of the procedure. The $m$ out of
$n$ bootstrap procedure is also considered and shown to be consistent. We also
provide sufficient conditions for any bootstrap procedure to be consistent in
this scenario.
