The Annals of Statistics

Nonparametric confidence intervals for monotone functions

Piet Groeneboom and Geurt Jongbloed

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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.

Article information

Ann. Statist., Volume 43, Number 5 (2015), 2019-2054.

Received: June 2014
Revised: January 2015
First available in Project Euclid: 3 August 2015

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G05: Estimation 62N01: Censored data models
Secondary: 62G20: Asymptotic properties

LR test MLE confidence intervals isotonic estimate smoothed MLE bootstrap


Groeneboom, Piet; Jongbloed, Geurt. Nonparametric confidence intervals for monotone functions. Ann. Statist. 43 (2015), no. 5, 2019--2054. doi:10.1214/15-AOS1335.

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