Moderate deviations and nonparametric inference for monotone functions
type: Item
instance of: scholarly article
title: Moderate deviations and nonparametric inference for monotone functions
zbMATH Open document ID: 1434.60087
DOI: 10.1214/17-AOS1583
author: Jie Xiong
author: Xingqiu Zhao
author: Fuqing Gao
published in: The Annals of Statistics
publication date: 2018-06-29T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.aos/1525313081
review text: ``This paper considers self-normalized limits and moderate deviations of nonparametric maximum likelihood estimators for monotone functions. We obtain their self-normalized Cramer-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.'' The paper is structured in 5 chapters: 1. Introduction; 2. Main results, 2.1 Current status data, 2.2 Grenander-type estimator, 2.2.1 The monotone density function, 2.2.2 Monotone hazard function with right-censored data, 2.2.3 The monotone regression model; 3. Proofs of the main results; 4. Applications, 4.1 Hypothesis testing, 4.2 Pointwise confidence interval, 4.3 Simulation study, 4.4 Conclusions; 5. Proofs of Theorems 3.1 and 3.2, 5.1 An equivalent representation of the moderate deviations, 5.2 A strong approximation theorem associated with the NPMLE (nonparametric maximum likelihood estimator), 5.3 Exponential equivalence, 5.4. Proof of Theorem 3.1, 5.5 Proof of Theorem 3.2; Acknowledgements; References (37 references); Supplementary material. For the supplementary material see \url{https://doi.org/10.1214/17-AOS1583SUPP} or \url{https://www.polyu.edu.hk/ama/staff/zhaoxq/Papers/Gao-Xiong-Zhao-AOS2018.pdf} The supplement [\textit{F. Gao} et al. [Ann. Stat. 46, No. 3, 1225--1254 (2018; Zbl 06897928)] contains all remaining technical proofs omitted from the main text due to space constraints, in which the authors prove the Lemmas 5.4 …
reviewed by: Ludwig Paditz
Mathematics Subject Classification ID: 60F10
Mathematics Subject Classification ID: 62G20
Mathematics Subject Classification ID: 62G07
zbMATH DE Number: 6897928
zbMATH Keywords: Grenander estimator
zbMATH Keywords: interval censored data
zbMATH Keywords: large deviations
zbMATH Keywords: moderate deviations
zbMATH Keywords: nonparametric MLE
zbMATH Keywords: self-normalized limit
zbMATH Keywords: strong approximation
zbMATH Keywords: Talagrand inequality
MaRDI profile type: Publication
cites work: On the Bootstrap of the Maximum Score Estimator
cites work: Likelihood based inference for monotone response models
cites work: Likelihood ratio tests for monotone functions.
cites work: Confidence Intervals for Current Status Data
cites work: Score Statistics for Current Status Data: Comparisons with Likelihood Ratio and Wald Statistics
cites work: Cramér-type moderate deviations for Studentized two-sample \(U\)-statistics with applications
cites work: Q4391441
cites work: Self-Normalized Processes
cites work: A law of the iterated logarithm for Grenander's estimator
cites work: Sharp asymptotics for isotonic regression
cites work: On the \(\mathbb L_p\)-error of monotonicity constrained estimators
cites work: The limit distribution of the \(L_{\infty}\)-error of Grenander-type estimators
cites work: Moderate deviations and large deviations for kernel density estimators
cites work: Delta method in large deviations and moderate deviations for estimators
cites work: On consistency of kernel density estimators for randomly censored data: Rates holding uniformly over adaptive intervals
cites work: Q3242523
cites work: Q5369130
cites work: Q3124804
cites work: Maximum smoothed likelihood estimators for the interval censoring model
cites work: Asymptotic normality of the \(L_1\) error of the Grenander estimator
cites work: Nonparametric Estimation under Shape Constraints
cites work: Nonparametric confidence intervals for monotone functions
cites work: Maximum smoothed likelihood estimation and smoothed maximum likelihood estimation in the current status model
cites work: Q4023149
cites work: An extremal limit theorem for the argmax process of Brownian motion minus a parabolic drift
cites work: Q4834282
cites work: Cramér-type moderate deviation for the maximum of the periodogram with application to simultaneous tests in gene expression time series
cites work: A Cramér moderate deviation theorem for Hotelling's \(T^{2}\)-statistic with applications to global tests
cites work: Q4825412
cites work: Q4085017
cites work: Q5572964
cites work: The statistical analysis of interval-censored failure time data.
cites work: New concentration inequalities in product spaces
cites work: Weak convergence and empirical processes. With applications to statistics
description: scientific article; zbMATH DE number 6897928