Bootstrap failure for a uniform endpoint ($P_{\ast }(X_{(n)}^{\ast }=X_{(n)})=1-(1-1/n{)}^n$)

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 $1-(1-1/n{)}^n\to 1-e^{-1}$. 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.