  In this paper, we investigate the (in)-consistency of different bootstrap
methods for constructing confidence intervals in the class of estimators that
converge at rate $n^{1/3}$. The Grenander estimator, the nonparametric maximum
likelihood estimator of an unknown nonincreasing density function $f$ on
$[0,\infty)$, is a prototypical example. We focus on this example and explore
different approaches to constructing bootstrap confidence intervals for
$f(t_0)$, where $t_0\in(0,\infty)$ is an interior point. We find that the
bootstrap estimate, when generating bootstrap samples from the empirical
distribution function $\mathbb{F}_n$ or its least concave majorant
$\tilde{F}_n$, does not have any weak limit in probability. We provide a set of
sufficient conditions for the consistency of any bootstrap method in this
example and show that bootstrapping from a smoothed version of $\tilde{F}_n$
leads to strongly consistent estimators. The $m$ out of $n$ bootstrap method is
also shown to be consistent while generating samples from $\mathbb{F}_n$ and
$\tilde{F}_n$.
