Inconsistency of bootstrap: The Grenander estimator

Abstract

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 $\mathbb{F}_{n}$ or its least concave majorant F̃_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 F̃_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 F̃_n.