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Estimating a distribution function subject to a stochastic order restriction: a comparative study

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  • Ori Davidov
  • George Iliopoulos

Abstract

In this article, we compare four nonparametric estimators of a distribution function (DF), estimated under a stochastic order restriction. The estimators are compared by simulation using four criteria: (1) the estimation of cumulative DFs; (2) the estimation of quantiles; (3) the estimation of moments and other functionals; and (4) as tools for testing for stochastic order. Our simulation study shows that estimators based on the pointwise maximum-likelihood estimator ( p -MLE) outperform all other estimators when the underlying distributions are 'close' to each other. The gain in efficiency may be as high as 25%. If the DFs are far apart then the p -MLE may not be the best. However, the efficiency loss using the p -MLE relative to the best estimator in each case is generally low (about 5%). We also find that the test based on the p -MLE is the most powerful in the majority of cases although the gain in power relative to other tests is generally small.

Suggested Citation

  • 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.
    • Handle: RePEc:taf:gnstxx:v:24:y:2012:i:4:p:923-933
    DOI: 10.1080/10485252.2012.710333
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    1. 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.
    2. Ori Davidov & Amir Herman, 2012. "Ordinal dominance curve based inference for stochastically ordered distributions," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 74(5), pages 825-847, November.
    3. Yongseok Park & Jeremy M. G. Taylor & John D. Kalbfleisch, 2012. "Pointwise nonparametric maximum likelihood estimator of stochastically ordered survivor functions," Biometrika, Biometrika Trust, vol. 99(2), pages 327-343.
    4. Hammou El Barmi & Hari Mukerjee, 2005. "Inferences Under a Stochastic Ordering Constraint: The k-Sample Case," Journal of the American Statistical Association, American Statistical Association, vol. 100, pages 252-261, March.
    5. Rojo, Javier & Samaniego, Francisco J., 1991. "On nonparametric maximum likelihood estimation of a distribution uniformly stochastically smaller than a standard," Statistics & Probability Letters, Elsevier, vol. 11(3), pages 267-271, March.
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