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Abstract:
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We consider the problem of bootstrapping the Grenander estimator, the nonparametric maximum likelihood estimator of $f$, a non-increasing density on positive real line. The Grenander estimator converges weakly to a non-normal limit involving nuisance parameters at $n^{1/3}$-rate. The non-standard rate of convergence makes the usual bootstrap procedures a suspect in this situation. In this paper we explore different bootstrap methods and investigate the consistency of the procedures. We show that the bootstrap statistic based on i.i.d. samples from the empirical distribution function, does not have any weak limit, conditional on the data, in probability. A similar phenomenon holds while bootstrapping from the least concave majorant of the empirical distribution. A suitable version of smoothed bootstrap achieves consistency, as does the $m$ out of $n$ bootstrap.
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