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.

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Last updated: 24 Mar 2023