  The goal of this paper is to study the bootstrap for the Grenander estimator.
The first result is a proof of the inconsistency of the nonparametric bootstrap
for the Grenander estimator at a given point. The second result is the
development and verification of a bootstrap for the $L_1$ confidence band for
the Grenander estimator. As part of this work, kernel estimators are studied as
alternatives to the Grenander estimator. We show that when the second
derivative of the true density is assumed to be uniformly bounded, there exist
kernel estimators with faster convergence rates than the Grenander estimator.
We study the implications of this in developing $L_1$ and uniform confidence
bands and discuss some open questions.
