Minimum distance estimation of the distributionfunctions of stochastically ordered random variables


Stochastic ordering of distributions can be a natural and minimal restriction in anestimation 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 distri-bution functions nearly satisfy the stochastic orderings, the NPMLE and the empirical cumulativedistribution functions may differ substantially. In many settings, this can make the NPMLE seemto be an unappealing estimator. As an alternative to the NPMLE, we propose a minimum dis-tance estimator of distribution functions subject to stochastic ordering constraints. Consistencyof the minimum distance estimator is proved, and superior performance is demonstrated througha simulation study. We demonstrate the use of the methodology to assess the reproducibility ofgradings of nuclear sclerosis from fundus photographs.

Journal of the Royal Statistical Society, Series C