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Generalized Fiducial Inference: A Review and New Results

Author

Listed:
  • Jan Hannig
  • Hari Iyer
  • Randy C. S. Lai
  • Thomas C. M. Lee

Abstract

R. A. Fisher, the father of modern statistics, proposed the idea of fiducial inference during the first half of the 20th century. While his proposal led to interesting methods for quantifying uncertainty, other prominent statisticians of the time did not accept Fisher’s approach as it became apparent that some of Fisher’s bold claims about the properties of fiducial distribution did not hold up for multi-parameter problems. Beginning around the year 2000, the authors and collaborators started to reinvestigate the idea of fiducial inference and discovered that Fisher’s approach, when properly generalized, would open doors to solve many important and difficult inference problems. They termed their generalization of Fisher’s idea as generalized fiducial inference (GFI). The main idea of GFI is to carefully transfer randomness from the data to the parameter space using an inverse of a data-generating equation without the use of Bayes’ theorem. The resulting generalized fiducial distribution (GFD) can then be used for inference. After more than a decade of investigations, the authors and collaborators have developed a unifying theory for GFI, and provided GFI solutions to many challenging practical problems in different fields of science and industry. Overall, they have demonstrated that GFI is a valid, useful, and promising approach for conducting statistical inference. The goal of this article is to deliver a timely and concise introduction to GFI, to present some of the latest results, as well as to list some related open research problems. It is authors’ hope that their contributions to GFI will stimulate the growth and usage of this exciting approach for statistical inference. Supplementary materials for this article are available online.

Suggested Citation

  • Jan Hannig & Hari Iyer & Randy C. S. Lai & Thomas C. M. Lee, 2016. "Generalized Fiducial Inference: A Review and New Results," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 111(515), pages 1346-1361, July.
  • Handle: RePEc:taf:jnlasa:v:111:y:2016:i:515:p:1346-1361
    DOI: 10.1080/01621459.2016.1165102
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    Citations

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    Cited by:

    1. Patrick Leung & Catherine S. Forbes & Gael M Martin & Brendan McCabe, 2019. "Forecasting Observables with Particle Filters: Any Filter Will Do!," Monash Econometrics and Business Statistics Working Papers 22/19, Monash University, Department of Econometrics and Business Statistics.
    2. Hsin-I Lee & Hungyen Chen & Hirohisa Kishino & Chen-Tuo Liao, 2016. "A Reference Population-Based Conformance Proportion," Journal of Agricultural, Biological and Environmental Statistics, Springer;The International Biometric Society;American Statistical Association, vol. 21(4), pages 684-697, December.
    3. La Vecchia, Davide & Moor, Alban & Scaillet, Olivier, 2023. "A higher-order correct fast moving-average bootstrap for dependent data," Journal of Econometrics, Elsevier, vol. 235(1), pages 65-81.
    4. Piero Veronese & Eugenio Melilli, 2021. "Confidence Distribution for the Ability Parameter of the Rasch Model," Psychometrika, Springer;The Psychometric Society, vol. 86(1), pages 131-166, March.
    5. Jan Hannig & Hari Iyer, 2022. "Testing for calibration discrepancy of reported likelihood ratios in forensic science," Journal of the Royal Statistical Society Series A, Royal Statistical Society, vol. 185(1), pages 267-301, January.
    6. Beau Coker & Cynthia Rudin & Gary King, 2021. "A Theory of Statistical Inference for Ensuring the Robustness of Scientific Results," Management Science, INFORMS, vol. 67(10), pages 6174-6197, October.
    7. Yixuan Zou & Jan Hannig & Derek S. Young, 2021. "Generalized fiducial inference on the mean of zero-inflated Poisson and Poisson hurdle models," Journal of Statistical Distributions and Applications, Springer, vol. 8(1), pages 1-15, December.
    8. K. Krishnamoorthy & Shanshan Lv, 2018. "Highest posterior mass prediction intervals for binomial and poisson distributions," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 81(7), pages 775-796, October.
    9. Gunnar Taraldsen, 2023. "The Confidence Density for Correlation," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 85(1), pages 600-616, February.
    10. Eugenio Melilli & Piero Veronese, 2024. "Confidence distributions and hypothesis testing," Statistical Papers, Springer, vol. 65(6), pages 3789-3820, August.
    11. Hezhi Lu & Hua Jin & Zhining Wang & Chao Chen & Ying Lu, 2019. "Prior-free probabilistic interval estimation for binomial proportion," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 28(2), pages 522-542, June.
    12. Shin-Fu Tsai, 2019. "Comparing Coefficients Across Subpopulations in Gaussian Mixture Regression Models," Journal of Agricultural, Biological and Environmental Statistics, Springer;The International Biometric Society;American Statistical Association, vol. 24(4), pages 610-633, December.
    13. Gunnar Taraldsen & Jarle Tufto & Bo H. Lindqvist, 2022. "Improper priors and improper posteriors," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 49(3), pages 969-991, September.
    14. Piao Chen & Zhi‐Sheng Ye & Xun Xiao, 2019. "Pairwise model discrimination with applications in lifetime distributions and degradation processes," Naval Research Logistics (NRL), John Wiley & Sons, vol. 66(8), pages 675-686, December.
    15. Seungyong Hwang & Randy C. S. Lai & Thomas C. M. Lee, 2022. "Generalized Fiducial Inference for Threshold Estimation in Dose–Response and Regression Settings," Journal of Agricultural, Biological and Environmental Statistics, Springer;The International Biometric Society;American Statistical Association, vol. 27(1), pages 109-124, March.
    16. Andrew Gelman & Christian Hennig, 2017. "Beyond subjective and objective in statistics," Journal of the Royal Statistical Society Series A, Royal Statistical Society, vol. 180(4), pages 967-1033, October.
    17. Veronese, Piero & Melilli, Eugenio, 2018. "Some asymptotic results for fiducial and confidence distributions," Statistics & Probability Letters, Elsevier, vol. 134(C), pages 98-105.
    18. Yufan Wang & Xingzhong Xu, 2023. "A Posterior p -Value for Homogeneity Testing of the Three-Sample Problem," Mathematics, MDPI, vol. 11(18), pages 1-25, September.
    19. Fröhlich, Andreas & Weng, Annegret, 2018. "Parameter uncertainty and reserve risk under Solvency II," Insurance: Mathematics and Economics, Elsevier, vol. 81(C), pages 130-141.
    20. Ionut Bebu & George Luta & Thomas Mathew & Brian K. Agan, 2016. "Generalized Confidence Intervals and Fiducial Intervals for Some Epidemiological Measures," IJERPH, MDPI, vol. 13(6), pages 1-13, June.
    21. Wu, Suofei & Hannig, Jan & Lee, Thomas C.M., 2022. "Uncertainty quantification for honest regression trees," Computational Statistics & Data Analysis, Elsevier, vol. 167(C).

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