The Annals of Statistics

Moderate deviations and nonparametric inference for monotone functions

Fuqing Gao, Jie Xiong, and Xingqiu Zhao

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This paper considers self-normalized limits and moderate deviations of nonparametric maximum likelihood estimators for monotone functions. We obtain their self-normalized Cramér-type moderate deviations and limit distribution theorems for the nonparametric maximum likelihood estimator in the current status model and the Grenander-type estimator. As applications of the results, we present a new procedure to construct asymptotical confidence intervals and asymptotical rejection regions of hypothesis testing for monotone functions. The theoretical results can guarantee that the new test has the probability of type II error tending to 0 exponentially. Simulation studies also show that the new nonparametric test works well for the most commonly used parametric survival functions such as exponential and Weibull survival distributions.

Article information

Ann. Statist., Volume 46, Number 3 (2018), 1225-1254.

Received: September 2016
Revised: March 2017
First available in Project Euclid: 3 May 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F10: Large deviations 62G20: Asymptotic properties
Secondary: 62G07: Density estimation

Grenander estimator interval censored data large deviations moderate deviations nonparametric MLE self-normalized limit strong approximation Talagrand inequality


Gao, Fuqing; Xiong, Jie; Zhao, Xingqiu. Moderate deviations and nonparametric inference for monotone functions. Ann. Statist. 46 (2018), no. 3, 1225--1254. doi:10.1214/17-AOS1583.

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Supplemental materials

  • Supplement to “Moderate deviations and nonparametric inference for monotone functions”. The supplement [Gao, Xiong and Zhao (2017)] contains all remaining technical proofs omitted from the main text due to space constraints, in which we prove Lemmas 5.4 and 5.5, Proposition 2.2, Theorem 2.3 and its Corollaries 2.1–2.3.