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Fiducial inference in the pivotal family of distributions

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Abstract

In this paper a family, called the pivotal family, of distributions is considered. A pivotal family is determined by a generalized pivotal model. Analytical results show that a great many parametric families of distributions are pivotal. In a pivotal family of distributions a general method of deriving fiducial distributions of parameters is proposed. In the method a fiducial model plays an important role. A fiducial model is a function of a random variable with a known distribution, called the pivotal random element, when the observation of a statistic is given. The method of this paper includes some other methods of deriving fiducial distributions. Specially the first fiducial distribution given by Fisher can be derived by the method. For the monotone likelihood ratio family of distributions, which is a pivotal family, the fiducial distributions have a frequentist property in the Neyman-Pearson view. Fiducial distributions of regular parametric functions also have the above frequentist property. Some advantages of the fiducial inference are exhibited in four applications of the fiducial distribution. Many examples are given, in which the fiducial distributions cannot be derived by the existing methods.

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References

  1. Fisher, R. A., Inverse probability, Proc. Cambridge Philos. Soc., 1930, 26: 528–535.

    MATH  Google Scholar 

  2. Fisher, R. A., Contributions to mathematical statistics, New York: John Wiley & Sons, Inc., 1950.

    Google Scholar 

  3. Fisher, R. A., Statistical methods and scientific inference, London: Oliver and Boyd, 1956.

    Google Scholar 

  4. Fraser, D. A. S., On fiducial inference, Ann. Math. Statist., 1961, 32: 661–671.

    MATH  MathSciNet  Google Scholar 

  5. Fraser, D. A. S., The fiducial method and invariance, Biometrika, 1961, 48: 261–280.

    MATH  MathSciNet  Google Scholar 

  6. Fraser, D. A. S., The structure of inference, New York: Wiley, 1968.

    Google Scholar 

  7. Hora, R. B., Buehler, R. J., Fiducial theory and invariant estimation, Ann. Math. Statist., 1966, 37: 643–656.

    MathSciNet  Google Scholar 

  8. Hora, R. B., Buehler, R. J., Fiducial theory and invariant prediction, Ann. Math. Statist., 1967, 38: 795–801.

    MathSciNet  Google Scholar 

  9. Dawid, A. P., Stone, M., The functional model basis of fiducial inference, The Annals of Statistics, 1982, 10: 1054–1067.

    MathSciNet  Google Scholar 

  10. Dawid, A. P., Wang, J., Fiducial prediction and semi-Bayesian inference, The Annals of Statistics, 1993, 21 1119–1138.

    MathSciNet  Google Scholar 

  11. Barnard, G. A., Pivotal models and the fiducial argument, International Statistical Review, 1995, 63: 309–323.

    MATH  Google Scholar 

  12. Lindley, D. V., Fiducial distributions and Bayes’ theorem, J. Roy. Statist. Soc. B, 1958, 20: 102–107.

    MATH  MathSciNet  Google Scholar 

  13. Fraser, D. A. S., On the consistency of the fiducial method, J. Roy. Statist. Soc. B, 1962, 24: 425–434.

    MATH  Google Scholar 

  14. Fraser, D. A. S., Letter to the editors, International Statistical Review, 1996, 64: 231–236.

    MATH  Google Scholar 

  15. Barnard, G. A., Some logical aspects of the fiducial argument, J. Roy. Statist. Soc. B, 1963, 25: 111–114.

    MATH  MathSciNet  Google Scholar 

  16. Barnard, G. A., Logical aspects of the fiducial argument, Bull. Int. Statist. Inst., 1963, 40: 870–883.

    MathSciNet  Google Scholar 

  17. Barnard, G. A., Pivotal inference and the Bayesian controversy, Bull. Int. Statist. Inst., 1977, 47: 543–551.

    MATH  MathSciNet  Google Scholar 

  18. Dawid, A. P., Stone, M., Zidek, J. V., Marginalizatin paradoxes in Bayesian and structural inference (with discussion), J. Roy. Statist. Soc. B, 1973, 35: 189–233.

    MathSciNet  Google Scholar 

  19. Wilkinson, D., On resolving the controversy in statistical inference (with discussion), J. Roy. Statist. Soc. Ser. B, 1977, 39: 119–171.

    MATH  MathSciNet  Google Scholar 

  20. Pedersen, J. G., Fiducial inference, International Statistical Review, 1978, 46: 147–170.

    MATH  MathSciNet  Google Scholar 

  21. Zabell, S. L., R. A. Fisher and fiducial argument, Statistical Science, 1992, 7: 369–387.

    MATH  MathSciNet  Google Scholar 

  22. Seidenfeld, T., R. A. Fisher’s fiducial argument and Bayes’ theorem, Statistical Science, 1992, 7: 358–368.

    MATH  MathSciNet  Google Scholar 

  23. Wang, Y. H., Fiducial intervals: what are they? The American Statistician, 2000, 54: 105–111.

    MathSciNet  Google Scholar 

  24. Efron, B., R. A. Fisher in the 21st Century, Statistical Science, 1998, 13: 95–122.

    MATH  MathSciNet  Google Scholar 

  25. Fisher, R. A., Two new properties of mathematical likelihood, Proc. Royal Soc. A, 1934, 144: 285–307.

    MATH  Google Scholar 

  26. Fisher, R. A., The fiducial argument in statistical inference, Ann. Eugenics, 1935, 6: 391–398.

    Google Scholar 

  27. Fraser, D. A. S., Necessary analysis and adaptive inference, J. Amer. Statist. Assoc., 1976, 71: 99–110.

    MATH  MathSciNet  Google Scholar 

  28. Blyth, C. R. and Bondar, J. V., A Neyman-Pearson-Wald view of fiducial probability, in Foundations of Statistical Inference (eds. MacNeill, I. B., Umphrey, G. J.), New York: D. Reidel Publishing Company, 1987, 9–20.

    Google Scholar 

  29. Kroese, A. H., Salmoé, D., Schaafsma, W., An optimum property for fiducial inference, Statistics & Decision 1999, 17: 285–291.

    MathSciNet  Google Scholar 

  30. Mauldon, J. D., Pivotal quantities for Wishart’s and related distributions and a paradox in fiducial theory, J. Roy. Statist. Soc. B, 1955, 17: 79–85.

    MathSciNet  Google Scholar 

  31. Stein, C., An example of wide discrepancy between fiducial and confidence intervals, Ann. Math. Statist., 1959, 30: 877–880.

    MATH  MathSciNet  Google Scholar 

  32. Grundy, P. M., Fiducial distributions and prior distributions: an example in which the former cannot be associated with the latter, J. R. Statist. Soc. B, 1956, 18: 217–221.

    MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Beijing Institute of Technology, Beijing, 100081, China

    Xu Xingzhong

  2. Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100080, China

    Li Guoying

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  1. Xu Xingzhong
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  2. Li Guoying
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Correspondence to Xu Xingzhong.

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Xu, X., Li, G. Fiducial inference in the pivotal family of distributions. SCI CHINA SER A 49, 410–432 (2006). https://doi.org/10.1007/s11425-006-0410-4

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  • DOI: https://doi.org/10.1007/s11425-006-0410-4

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