Bootstrapping for threshold standard error

In my 2003 Ecological Modelling paper on ecological threshold, I used the bootstrapping method for estimating the standard error of a change point variable.  The change point was estimated as the binary break point that resulted in the greatest reduction in deviance.  Ian McKeague of FSU (now at Columbia) told me about Buhlmann and Yu (2002), which suggested that bootstrapping is inappropriate for a change point problem.  I was convinced that the bootstrap confidence interval is wrong, but was never able to explain the reasons convincingly until recently.

A change point problem limits the potential break points to be the gaps (intervals between distinct values) in the given data set.  These potential break points do not change from bootstrap sample to bootstrap sample, only that some gaps may become wider.  Should the population were sampled, the potential number of break points is infinite.  As a result, bootstrap estimated break point is of an artificially smaller sampling variance than the variance of the true sampling distribution of the break point. This reduced sampling variation is likely the cause of a much smaller standard error or narrower confidence interval when estimated using bootstrapping.  In Banerjeer and McKeague (2007), a simulation is performed to show that a 95% confidence interval based on bootstrapping covers the underlying true change point far less than 95% of the time.