Open Access
2017 Weak convergence of the least concave majorant of estimators for a concave distribution function
Brendan K. Beare, Zheng Fang
Electron. J. Statist. 11(2): 3841-3870 (2017). DOI: 10.1214/17-EJS1349


We study the asymptotic behavior of the least concave majorant of an estimator of a concave distribution function under general conditions. The true concave distribution function is permitted to violate strict concavity, so that the empirical distribution function and its least concave majorant are not asymptotically equivalent. Our results are proved by demonstrating the Hadamard directional differentiability of the least concave majorant operator. Standard approaches to bootstrapping fail to deliver valid inference when the true distribution function is not strictly concave. While the rescaled bootstrap of Dümbgen delivers asymptotically valid inference, its performance in small samples can be poor, and depends upon the selection of a tuning parameter. We show that two alternative bootstrap procedures—one obtained by approximating a conservative upper bound, the other by resampling from the Grenander estimator—can be used to construct reliable confidence bands for the true distribution. Some related results on isotonic regression are provided.


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Brendan K. Beare. Zheng Fang. "Weak convergence of the least concave majorant of estimators for a concave distribution function." Electron. J. Statist. 11 (2) 3841 - 3870, 2017.


Received: 1 October 2016; Published: 2017
First available in Project Euclid: 18 October 2017

zbMATH: 06796557
MathSciNet: MR3714300
Digital Object Identifier: 10.1214/17-EJS1349

Primary: 62G09 , 62G20

Keywords: Grenander estimator , Hadamard directional derivative , least concave majorant , rescaled bootstrap

Vol.11 • No. 2 • 2017
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