You bootstrap the sample maximum of n = 100 draws from Uniform(0, theta) to build a confidence interval for theta. Give the probability that a bootstrap resample reproduces the observed maximum, and explain why the resulting interval has the wrong coverage even as n grows.

You bootstrap the sample maximum of n = 100 draws from Uniform(0, theta) to build a confidence interval for theta. Give the probability that a bootstrap resample reproduces the observed maximum, and explain why the resulting interval has the wrong coverage even as n grows.

Approach: Compute the chance the observed maximum is missed by all n resampled draws, then compare the atom this creates with the continuous limit law of the true statistic.

0.634. A bootstrap resample misses the observed maximum only if all 100 draws avoid it, with probability (1 - 1/100)^100 = 0.366, so the bootstrap maximum equals the sample maximum with probability 0.634, and the limiting value for any large n is 1 - e^{-1} = 0.632. The bootstrap distribution of M* - M therefore carries an atom of mass 0.632 at zero, while the true limit law is continuous, since n*(theta - M)/theta converges to an Exponential(1). No amount of data removes the atom, so the bootstrap is inconsistent here and the interval built from it has coverage that does not approach 95%. The failure is structural: the bootstrap approximates the sampling distribution by resampling from the empirical distribution, whose support ends exactly at M, so it can never place mass above the observed maximum. Consistency needs the statistic to be a smooth functional of the distribution, and extreme values are not. Use the exact pivot or an m-out-of-n bootstrap with m/n tending to zero.

Follow-up: For which of the sample median, the sample variance and the number of distinct values does the ordinary bootstrap remain consistent, and why?

Key concepts: bootstrap consistency, empirical distribution, extreme value, sampling distribution.