# Is there any intuition as to why the bootstrap percentile confidence interval has coverage error?

When a parameter's sampling distribution is skewed, the bootstrap percentile confidence has coverage error (i.e. a 95% percent confidence interval is not really 95%).

See, for example, Bootstrap confidence intervals: when, which, what? A practical guide for medical statisticians by Carpenter and Bithell.

Is there any intuitive explanation why this is the case?

• The quick answer is that at its heart it relies on there being a reasonably good Normal approximation to the distribution of a monotonic transform of the statistic, and such a transform may not exist. Alternatively, you can think of it as representing the fact that, e.g., the $95^{th}$ bootstrap percentile of a sample of size $n$ from some distributions may be a seriously biased estimator of the $95^{th}$ percentile of the population (the bias decreases at $O(n^{-1/2})$ as $n \to \infty$.) – jbowman Feb 20 '18 at 17:31
• Sample size of n refers to the number of observations and not the number of bootstrap repetitions, correct? – Great38 Feb 20 '18 at 17:47
• Correct, increasing the number of bootstrap samples doesn't help with this particular problem. – jbowman Feb 20 '18 at 17:58