Pointwise nonparametric maximum likelihood estimator of stochastically ordered survivor functions

• Published in: Biometrika 2012-06, Vol.99 (2), p.327-343
• Main Authors: PARK, YONGSEOK, TAYLOR, JEREMY M. G., KALBFLEISCH, JOHN D.
• Format: Article
• Language: English
• Subjects: Algorithms; Applications; Asymptotic methods; Biology, psychology, social sciences; Bootstrap resampling; Censorship; Confidence interval; Data processing; Estimation methods; Estimators; Exact sciences and technology; General topics; Kaplan Meier estimator; Mathematical functions; Mathematics; Maximum likelihood estimation; Maximum likelihood estimators; Maximum likelihood method; Multivariate analysis; Nonparametric inference; Parameter estimation; Probability and statistics; Prostate cancer; Reliability functions; Sample size; Sciences and techniques of general use; Statistics; Stochastic models; Stochasticity; Studies
• ISSN: 0006-3444; 1464-3510; 1464-3510; 0006-3444
• DOI: 10.1093/biomet/ass006

In this paper, we consider estimation of survivor functions from groups of observations with right-censored data when the groups are subject to a stochastic ordering constraint. Many methods and algorithms have been proposed to estimate distribution functions under such restrictions, but none have completely satisfactory properties when the observations are censored. We propose a pointwise constrained nonparametric maximum likelihood estimator, which is defined at each time t by the estimates of the survivor functions subject to constraints applied at time t only. We also propose an efficient method to obtain the estimator. The estimator of each constrained survivor function is shown to be nonincreasing in t, and its consistency and asymptotic distribution are established. A simulation study suggests better small and large sample properties than for alternative estimators. An example using prostate cancer data illustrates the method.