Bootstrapping the Grenander estimator

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.