= Bootstrap failure for a uniform endpoint
{title2=$P_*(X_{(n)}^*=X_{(n)})=1-(1-1/n)^n$}
For a continuous <uniform distribution> on $[0,\theta]$, the scaled maximum gap $n(\theta-X_{(n)})/\theta$ tends to a unit-rate <exponential distribution>. An ordinary <bootstrap sample> includes the unique observed maximum with probability $1-(1-1/n)^n\to1-e^{-1}$. The bootstrap maximum gap therefore retains an atom at zero, whereas the target limit is continuous. This proves failure of the ordinary empirical bootstrap for this endpoint functional; consistency of the endpoint estimate alone does not justify resampling its extreme-value limit.
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