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Minimum distance estimation of the distribution functions of stochastically ordered random variables
Author(s) -
Gang Ronald E.,
King William N.
Publication year - 2002
Publication title -
journal of the royal statistical society: series c (applied statistics)
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.205
H-Index - 72
eISSN - 1467-9876
pISSN - 0035-9254
DOI - 10.1111/1467-9876.00282
Subject(s) - mathematics , estimator , consistency (knowledge bases) , distribution (mathematics) , cumulative distribution function , empirical distribution function , statistics , probability density function , mathematical analysis , discrete mathematics
Summary. 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.