The Annals of Statistics
- Ann. Statist.
- Volume 38, Number 4 (2010), 1953-1977.
Inconsistency of bootstrap: The Grenander estimator
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 or its least concave majorant F̃n, 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 F̃n leads to strongly consistent estimators. The m out of n bootstrap method is also shown to be consistent while generating samples from and F̃n.
Ann. Statist., Volume 38, Number 4 (2010), 1953-1977.
First available in Project Euclid: 11 July 2010
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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. https://projecteuclid.org/euclid.aos/1278861239