Moderate deviations and nonparametric inference for monotone functions
``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 and 5.5, Proposition 2.2, Theorem 2.3 and its Corollaries 2.12.3. The supplement is structured in Appendix A with A.1. The proof of Lemmas 5.4 and 5.5, A.2 The proofs of the results for Grenander-type estimator and Appendix B with B.1 Large deviations, B.2 Talagrand inequality, B.3 KMT approximation theorem and 11 references, followed by R-codes for the tables 1--4 in the considered paper in 4.3 Simulation study. Appendix B contains some useful known results for the convenience of the reader.
scientific article; zbMATH DE number 6897928
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