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For $0\le t\le \theta$, independence and the uniform distribution give $P(X_{(n)}\le t)=(t/\theta {)}^n$. Consequently, for fixed $x\ge 0$ and $n>x$,For $x<0$ the survival probability is one. Thus the scaled endpoint gap converges in distribution to a unit-rate exponential law; it is not exactly exponential at finite $n$.

The plug-in estimator principle replaces the unknown sampling distribution by its empirical distribution and evaluates its population functional at that fitted distribution. Here the fitted support endpoint is $\hat{\theta }=X_{(n)}$, also the maximum-likelihood estimate: the likelihood is ${\theta }^{-n}{1}_{\{\theta \ge X_{(n)}\}}$ and decreases over its allowed support. For an empirical bootstrap sample $Y_1,\dots ,Y_n$, the corresponding gap isAlmost surely the original maximum is unique. The bootstrap gap equals zero precisely when this observation is selected at least once, soThe true limiting exponential distribution has no atom at zero. This nonvanishing discrepancy proves bootstrap failure for a uniform endpoint: the bootstrap distribution cannot converge to the correct limit, even though the endpoint estimate itself is consistent. It fails to recreate the unsampled population tail beyond the observed maximum. A parametric bootstrap drawing from $U[0,X_{(n)}]$, rather than from the discrete empirical sample, reproduces the scaled-gap law here.