Minimum distance estimation of the distribution functions of stochastically ordered random variables

• Published in: Applied statistics Vol. 51; no. 4; pp. 485 - 492
• Main Authors: Gangnon, Ronald E., King, William N.
• Format: Journal Article
• Language: English
• Published: Oxford, UK Blackwell Publishers 01.01.2002; Blackwell; Royal Statistical Society
• Series: Journal of the Royal Statistical Society Series C
• Subjects: Cramér-von Mises distance; Cumulative distribution functions; Data analysis; Distribution functions; Empirical distribution function; Error; Estimation bias; Estimators; Exact sciences and technology; Kaplan Meier estimator; Mathematical minima; Mathematics; Maximum likelihood estimation; Maximum likelihood estimators; Measurement; Measurement error; Minimum distance estimation; Nonparametric inference; Probability and statistics; Random variables; Sciences and techniques of general use; Sclerosis; Statistical analysis; Statistical methods; Statistics; Stochastic ordering constraints; Stochastic processes; Kaplan Meier estimator; Minimum; Empirical method; Stochastic order; Order statistic; Non parametric estimation; Empirical distribution function; Statistical estimation; Random distribution; Stochastic process; Statistical simulation; Cramer von Mises test; Function estimation; Medical science; Distance estimation; Consistent estimator; Distribution function; Minimal distance; Random function; Maximum likelihood; Random variable; Distance; Measurement error; Biased estimation
• ISSN: 0035-9254; 1467-9876
• DOI: 10.1111/1467-9876.00282

Stochastic ordering of distributions can be a natural and minimal restriction in an estimation problem. Such restrictions occur naturally in several settings in medical research. The standard estimator in such settings is the nonparametric maximum likelihood estimator (NPMLE). The NPMLE is known to be biased, and, even when the empirical cumulative distribution functions nearly satisfy the stochastic orderings, the NPMLE and the empirical cumulative distribution functions may differ substantially. In many settings, this can make the NPMLE seem to be an unappealing estimator. As an alternative to the NPMLE, we propose a minimum distance estimator of distribution functions subject to stochastic ordering constraints. Consistency of the minimum distance estimator is proved, and superior performance is demonstrated through a simulation study. We demonstrate the use of the methodology to assess the reproducibility of gradings of nuclear sclerosis from fundus photographs.