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Title: Inconsistency of the Bootstrap in Problems Exhibiting Cube Root Asymptotics
Speaker: Dr. Moulinath Banerjee University of Michigan
Date: 13 August 2008
Time: 4.00 p.m.
Venue: Lecture Hall - I, Dept. of Mathematics

We investigate the (in)-consistency of different bootstrap methods for constructing confidence intervals in the class of estimators that converge at rate $n^{1\\over 3}$. The Grenander estimator, the nonparametric maximum likelihood estimator of an unknown non-increasing 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 or its least concave majorant, does not have any weak limit in probability. Bootstrapping from a smoothed version of the least concave majorant, however, leads to strongly consistent estimators and the $m$ out of $n$ bootstrap method is also consistent. Our results cast serious doubt on some previous claims about bootstrap consistency (in the class of cube root problems) in the published literature.

Contact: +91 (80) 2293 2711, +91 (80) 2293 2265 ;     E-mail: chair.math[at]iisc[dot]ac[dot]in
Last updated: 24 Mar 2023