INCONSISTENCY OF BOOTSTRAP: THE GRENANDER ESTIMATOR

• 成果类型: Article
• 署名作者: Sen, Bodhisattva; Banerjee, Moulinath; Woodroofe, Michael
• 署名单位: Columbia University; University of Michigan System; University of Michigan
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/09-AOS777
• 发表日期: 2010
• 页码: 1953-1977
• 关键词: cube root asymptotics

摘要

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, infinity), is a prototypical example. We focus on this example and explore different approaches to constructing bootstrap confidence intervals for f(t(0)), where t(o) is an element of (0, infinity) is an interior point. We find that the bootstrap estimate, when generating bootstrap samples from the empirical distribution function F(n) or its least concave majorant (F) over tilde (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 (F) over tilde (n) leads to strongly consistent estimators. The m out of n bootstrap method is also shown to be consistent while generating samples from F(n) and (F) over tilde (n).