Inconsistency of bootstrap: The Grenander estimator

TL;DR

This paper examines the inconsistency issues of bootstrap methods for the Grenander estimator, a nonparametric estimator of a decreasing density, and identifies conditions under which bootstrap confidence intervals are consistent.

Contribution

It provides a detailed analysis of bootstrap inconsistency for the Grenander estimator and proposes conditions and methods for achieving bootstrap consistency.

Findings

Bootstrap estimates lack weak limits when sampling from empirical distribution functions.

Bootstrapping from smoothed versions of the estimator yields consistent confidence intervals.

The $m$ out of $n$ bootstrap method is consistent under certain sampling schemes.

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 $[0,\infty)$, 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 $\tilde{F}_n$.