The Annals of Statistics

Inconsistency of bootstrap: The Grenander estimator

Bodhisattva Sen, Moulinath Banerjee, and Michael Woodroofe

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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 n1/3. 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 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 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 n.

Article information

Ann. Statist., Volume 38, Number 4 (2010), 1953-1977.

First available in Project Euclid: 11 July 2010

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G09: Resampling methods 62G20: Asymptotic properties
Secondary: 62G07: Density estimation

Decreasing density empirical distribution function least concave majorant m out of n bootstrap nonparametric maximum likelihood estimate smoothed bootstrap


Sen, Bodhisattva; Banerjee, Moulinath; Woodroofe, Michael. Inconsistency of bootstrap: The Grenander estimator. Ann. Statist. 38 (2010), no. 4, 1953--1977. doi:10.1214/09-AOS777.

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