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August 2010 Inconsistency of bootstrap: The Grenander estimator
Bodhisattva Sen, Moulinath Banerjee, Michael Woodroofe
Ann. Statist. 38(4): 1953-1977 (August 2010). DOI: 10.1214/09-AOS777


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.


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Bodhisattva Sen. Moulinath Banerjee. Michael Woodroofe. "Inconsistency of bootstrap: The Grenander estimator." Ann. Statist. 38 (4) 1953 - 1977, August 2010.


Published: August 2010
First available in Project Euclid: 11 July 2010

zbMATH: 1202.62057
MathSciNet: MR2676880
Digital Object Identifier: 10.1214/09-AOS777

Primary: 62G09 , 62G20
Secondary: 62G07

Keywords: decreasing density , Empirical distribution function , least concave majorant , m out of n bootstrap , nonparametric maximum likelihood estimate , smoothed bootstrap

Rights: Copyright © 2010 Institute of Mathematical Statistics


Vol.38 • No. 4 • August 2010
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