Inconsistency of Bootstrap: the Grenander Estimator

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,∞), is a prototypical example. We focus on this example and explore different approaches to constructing bootstrap confidence intervals for f(t0), where t0 ∈ (0,∞) is an interior point. We find that the bootstrap estimate, when generating bootstrap samples from the empirical distribution function Fn or its least concave majorant F̃n, 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 F̃n leads to strongly consistent estimators. The m out of n bootstrap method is also shown to be consistent while generating samples from Fn and F̃n.