## Electronic Journal of Statistics

### Weak convergence of the least concave majorant of estimators for a concave distribution function

#### Abstract

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.

#### Article information

Source
Electron. J. Statist., Volume 11, Number 2 (2017), 3841-3870.

Dates
First available in Project Euclid: 18 October 2017

https://projecteuclid.org/euclid.ejs/1508292528

Digital Object Identifier
doi:10.1214/17-EJS1349

Mathematical Reviews number (MathSciNet)
MR3714300

Zentralblatt MATH identifier
06796557

Subjects
Primary: 62G09: Resampling methods 62G20: Asymptotic properties

#### Citation

Beare, Brendan K.; Fang, Zheng. Weak convergence of the least concave majorant of estimators for a concave distribution function. Electron. J. Statist. 11 (2017), no. 2, 3841--3870. doi:10.1214/17-EJS1349. https://projecteuclid.org/euclid.ejs/1508292528

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