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

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  • Ronald E. Gangnon
  • William N. King

Abstract

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.

Suggested Citation

  • Ronald E. Gangnon & William N. King, 2002. "Minimum distance estimation of the distribution functions of stochastically ordered random variables," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 51(4), pages 485-492, October.
    • Handle: RePEc:bla:jorssc:v:51:y:2002:i:4:p:485-492
    DOI: 10.1111/1467-9876.00282
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    Cited by:

    1. Darinka Dentcheva & Andrzej Ruszczynski, 2004. "Convexification of Stochastic Ordering," GE, Growth, Math methods 0402005, University Library of Munich, Germany, revised 05 Aug 2005.
    2. Darinka Dentcheva & Andrzej Ruszczynski, 2004. "Optimization Under First Order Stochastic Dominance Constraints," GE, Growth, Math methods 0403002, University Library of Munich, Germany, revised 07 Aug 2005.
    3. Karabatsos, George & Walker, Stephen G., 2007. "Bayesian nonparametric inference of stochastically ordered distributions, with Pólya trees and Bernstein polynomials," Statistics & Probability Letters, Elsevier, vol. 77(9), pages 907-913, May.
    4. Ori Davidov & George Iliopoulos, 2012. "Estimating a distribution function subject to a stochastic order restriction: a comparative study," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 24(4), pages 923-933, December.

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