## The Annals of Statistics

### Moderate deviations and nonparametric inference for monotone functions

#### Abstract

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

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

Dates
Revised: March 2017
First available in Project Euclid: 3 May 2018

https://projecteuclid.org/euclid.aos/1525313081

Digital Object Identifier
doi:10.1214/17-AOS1583

Mathematical Reviews number (MathSciNet)
MR3798002

Zentralblatt MATH identifier
06897928

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

#### Citation

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. https://projecteuclid.org/euclid.aos/1525313081

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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.