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

Published in at http://dx.doi.org/10.1214/09-AOS777 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (2)

• Bootstrapping the Grenander estimator
• A Kiefer--Wolfowitz comparison theorem for Wicksell's problem

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