Inconsistency of bootstrap: The Grenander estimator

By Bodhisattva Sen, Moulinath Banerjee, Michael Woodroofe. 2010

116 citations recorded by the source corpus.

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 $[0,\infty)$, 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 $\mathbb{F}_n$ or its least concave majorant $\tilde{F}_n$, 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 $\tilde{F}_n$ leads to strongly consistent estimators. The $m$ out of $n$ bootstrap method is also shown to be consistent while generating samples from $\mathbb{F}_n$ and $\tilde{F}_n$.

Subfields: Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/9129d0a432717e18ce75dee0241213ab2000c224

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Citations in the corpus

Testing equality of functions under monotonicity constraints (2012, methodology): “From the results in Sen et al. (2010) and Kosorok (2008), it appears that bootstrapping from the Grenander does not work when the statistic of interest is f̂n0 at a fixed point.”
Nonparametric Shape-Restricted Regression (2017, background): “It is well-known that both the above bootstrap schemes — bootstrapping pairs and bootstrapping residuals with f̌n = f̂n — yield inconsistent estimators of Hn; see [1, 56, 80, 119, 122].”
Regression diagnostics meets forecast evaluation: conditional calibration, reliability diagrams, and coefficient of determination (2021, background): “…(2020), and can be leveraged, though with hurdles, as rates of convergence depend on distributional assumptions and limit distributions involve nuisance parameters that need to be estimated, whereas the use of bootstrap methods might be impacted by the issues described by Sen et al. (2010).”