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October 2015 Nonparametric confidence intervals for monotone functions
Piet Groeneboom, Geurt Jongbloed
Ann. Statist. 43(5): 2019-2054 (October 2015). DOI: 10.1214/15-AOS1335


We study nonparametric isotonic confidence intervals for monotone functions. In [Ann. Statist. 29 (2001) 1699–1731], pointwise confidence intervals, based on likelihood ratio tests using the restricted and unrestricted MLE in the current status model, are introduced. We extend the method to the treatment of other models with monotone functions, and demonstrate our method with a new proof of the results of Banerjee–Wellner [Ann. Statist. 29 (2001) 1699–1731] and also by constructing confidence intervals for monotone densities, for which a theory remained be developed. For the latter model we prove that the limit distribution of the LR test under the null hypothesis is the same as in the current status model. We compare the confidence intervals, so obtained, with confidence intervals using the smoothed maximum likelihood estimator (SMLE), using bootstrap methods. The “Lagrange-modified” cusum diagrams, developed here, are an essential tool both for the computation of the restricted MLEs and for the development of the theory for the confidence intervals, based on the LR tests.


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Piet Groeneboom. Geurt Jongbloed. "Nonparametric confidence intervals for monotone functions." Ann. Statist. 43 (5) 2019 - 2054, October 2015.


Received: 1 June 2014; Revised: 1 January 2015; Published: October 2015
First available in Project Euclid: 3 August 2015

zbMATH: 1323.62040
MathSciNet: MR3375875
Digital Object Identifier: 10.1214/15-AOS1335

Primary: 62G05 , 62N01
Secondary: 62G20

Keywords: bootstrap , confidence intervals , isotonic estimate , LR test , MLE , smoothed MLE

Rights: Copyright © 2015 Institute of Mathematical Statistics


Vol.43 • No. 5 • October 2015
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