Generalized Fiducial Inference: A Review and New Results

Journal of the American Statistical Association

Volumn 111, Issue 515, 2016, Pages 1346-1361

  Hannig, Jan ^a   Iyer, Hari ^b   Lai, Randy C S ^c,d   Lee, Thomas C M ^c  

^a University of North Carolina at Chapel Hill   (United States)

^b NATIONAL INSTITUTE OF STANDARDS AND TECHNOLOGY   (United States)

^c UNIVERSITY OF CALIFORNIA   (United States)

^d UNIVERSITY OF MAINE   (United States)

Author keywords

Approximate Bayesian computations; Data generating equation; Fiducial inference; Jacobian calculation; Model selection; Uncertainty quantification

Indexed keywords

EID: 84991710967     PISSN: 01621459     EISSN: 1537274X     Source Type: Journal    
DOI: 10.1080/01621459.2016.1165102     Document Type: Review

References (106)

1. G.A.Barnard, (1995), “Pivotal Models and the Fiducial Argument,” International Statistical Reviews, 63, 309–323.
2. M.J.Bayarri, J.O.Berger, A.Forte, and G.García-Donato, (2012), “Criteria for Bayesian Model Choice With Application to Variable Selection,” The Annals of Statistics, 40, 1550–1577.
3. M.A.Beaumont, W.Zhang, and D.J.Balding, (2002), “Approximate Bayesian Computation in Population Genetics,” Genetics, 162, 2025–2035.
4. J.O.Berger, (1992), “On the Development of Reference Priors” (with discussion), Bayesian Statistics, 4, 35–60.
5. J.O.Berger, J.M.Bernardo, and D.Sun, (2009), “The Formal Definition of Reference Priors,” The Annals of Statistics, 37, 905–938.
6. ——— (2012), “Objective Priors for Discrete Parameter Spaces,” Journal of the American Statistical Association, 107, 636–648.
7. J.O.Berger, and L.R.Pericchi, (1996), “The Intrinsic Bayes Factor for Model Selection and Prediction,” Journal of the American Statistical Association, 91, 109–122.
8. ——— (2001), “Objective Bayesian Methods for Model Selection:Introduction and Comparison,” in Model Selection (IMS Lecture Notes Monogr. Ser., Vol. 38), ed. P. Lahiri, Beachwood, OH:Inst. Math. Statist., pp. 135–207.
9. J.O.Berger, and D.Sun, (2008), “Objective Priors for the Bivariate Normal Model,” The Annals of Statistics, 36, 963–982.
10. A.Birnbaum, (1961), “On the Foundations of Statistical Inference:Binary Experiments,” The Annals of Mathematical Statistics, 32, 414–435.
11. ——— (1962), “On the Foundations of Statistical Inference,” Journal of the American Statistical Association, 57, 269–326.
12. G.Blom, (1976), “Some Properties of Incomplete U-Statistics,” Biometrika, 63, 573–580.
13. E.Candes, and T.Tao, (2007), “The Dantzig Selector:Statistical Estimation When p is Much Larger Than n,” The Annals of Statistics, 35, 2313–2351.
14. G.Casella, and R.L.Berger, (2002), Statistical Inference (2nd ed.), Pacific Grove, CA:Wadsworth and Brooks/Cole Advanced Books and Software.
15. J.Chen, and Z.Chen, (2008), “Extended Bayesian Information Criteria for Model Selection With Large Model Spaces,” Biometrika, 95, 759–771.
16. A.K.L.Chiang, (2001), “A Simple General Method for Constructing Confidence Intervals for Functions of Variance Components,” Technometrics, 43, 356–367.
17. J.Cisewski, and J.Hannig, (2012), “Generalized Fiducial Inference for Normal Linear Mixed Models,” The Annals of Statistics, 40, 2102–2127.
18. A.P.Dawid, and M.Stone, (1982), “The Functional-Model Basis of Fiducial Inference,” The Annals of Statistics, 10, 1054–1074.
19. A.P.Dawid, M.Stone, and J.V.Zidek, (1973), “Marginalization Paradoxes in Bayesian and Structural Inference,” Journal of the Royal Statistical Society, Series B, 35, 189–233.
20. P.Del Moral, A.Doucet, and A.Jasra, (2006), “Sequential Monte Carlo Samplers,” Journal of the Royal Statistical Society, Series B, 68, 411–436.
21. A.P.Dempster, (1966), “New Methods for Reasoning Towards Posterior Distributions Based on Sample Data,” The Annals of Mathematical Statistics, 37, 355–374.
22. ——— (1968), “A Generalization of Bayesian Inference” (with discussion), Journal of the Royal Statistical Society, Series B, 30, 205–247.
23. ——— (2008), “The Dempster-Shafer Calculus for Statisticians,” International Journal of Approximate Reasoning, 48, 365–377.
24. R.Douc, and E.Moulines, (2008), “Limit Theorems for Weighted Samples With Applications to Sequential Monte Carlo Methods,” The Annals of Statistics, 36, 2344–2376.
25. A.Doucet, N.De Freitas, and N.Gordon, (2001), Sequential Monte Carlo Methods in Practice, New York:Springer.
26. L.E, J.Hannig, and H.K.Iyer, (2008), “Fiducial Intervals for Variance Components in an Unbalanced Two-Component Normal Mixed Linear Model,” Journal of the American Statistical Association, 103, 854–865.
27. ——— (2009), “Applications of Generalized Fiducial Inference,” Ph.D. Dissertation, Colorado State University, Fort Collins, CO.
28. P.T.Edlefsen, C.Liu, and A.P.Dempster, (2009), “Estimating Limits From Poisson Counting Data Using Dempster–Shafer Analysis,” The Annals of Applied Statistics, 3, 764–790.
29. B.Efron, (1998), “R.A. Fisher in the 21st Century,” Statistical Science, 13, 95–122.
30. J.Fan, and J.Lv, (2008), “Sure Independence Screening for Ultrahigh Dimensional Feature Space,” Journal of the Royal Statistical Society, Series B, 70, 849–911.
31. R.A.Fisher, (1922), “On the Mathematical Foundations of Theoretical Statistics,” Philosophical Transactions of the Royal Society of London, Series A, 222, 309–368.
32. ——— (1925), “Theory of Statistical Estimation,” Proceedings of the Cambridge Philosophical Society, 22, 700–725.
33. ——— (1930), “Inverse Probability,” Proceedings of the Cambridge Philosophical Society, xxvi, 528–535.
34. ——— (1933), “The Concepts of Inverse Probability and Fiducial Probability Referring to Unknown Parameters,” Proceedings of the Royal Society of London, Series A, 139, 343–348.
35. ——— (1935), “The Fiducial Argument in Statistical Inference,” The Annals of Eugenics, VI, 91–98.
36. A.M.Fraser, D.A.S.Fraser, and A.-M.Staicu, (2009), “The Second Order Ancillary:A Differential View With Continuity,” Bernoulli. Official Journal of the Bernoulli Society for Mathematical Statistics and Probability, 16, 1208–1223.
37. D.Fraser, and A.Naderi, (2008), “Exponential Models:Approximations for Probabilities,” Biometrika, 94, 1–9.
38. D.Fraser, N.Reid, and A.Wong, (2005), “What a Model With Data Says About Theta,” International Journal of Statistical Science, 3, 163–178.
39. D.A.S.Fraser, (1961a), “On Fiducial Inference,” The Annals of Mathematical Statistics, 32, 661–676.
40. ——— (1961b), “The Fiducial Method and Invariance,” Biometrika, 48, 261–280.
41. ——— (1966), “Structural Probability and a Generalization,” Biometrika, 53, 1–9.
42. ——— (1968), The Structure of Inference, New York-London-Sydney:Wiley.
43. ——— (2004), “Ancillaries and Conditional Inference,” Statistical Science, 19, 333–369.
44. ——— (2011), “Is Bayes Posterior Just Quick and Dirty Confidence?” Statistical Science, 26, 299–316.
45. Fraser, D. A. S., N.Reid, E.Marras, and G.Y.Yi, (2010), “Default Priors for Bayesian and Frequentist Inference,” Journal of the Royal Statistical Society, Series B, 72.
46. Y.S.Glagovskiy, (2006), “Construction of Fiducial Confidence Intervals For the Mixture of Cauchy and Normal Distributions,”Master’s Thesis, Department of Statistics, Colorado State University, Fort Collins, CO.
47. J.Hannig, (2009), “On Generalized Fiducial Inference,” Statistica Sinica, 19, 491–544.
48. ——— (2013), “Generalized Fiducial Inference via Discretization,” Statistica Sinica, 23, 489–514.
49. ——— (2014), Discussion of “On the Birnbaum Argument for the Strong Likelihood Principle” by D. G. Mayo, Statistical Science, 29, 254–258.
50. J.Hannig, L.E, A.Abdel-Karim, and H.K.Iyer, (2006a), “Simultaneous Fiducial Generalized Confidence Intervals for Ratios of Means of Lognormal Distributions,” Austrian Journal of Statistics, 35, 261–269.
51. J.Hannig, H.K.Iyer, and P.Patterson, (2006b), “Fiducial Generalized Confidence Intervals,” Journal of American Statistical Association, 101, 254–269.
52. J.Hannig, H.K.Iyer, and J.C.-M.Wang, (2007), “Fiducial Approach to Uncertainty Assessment:Accounting for Error Due to Instrument Resolution,” Metrologia, 44, 476–483.
53. J.Hannig, R.C.S.Lai, and T.C.M.Lee, (2014), “Computational Issues of Generalized Fiducial Inference,” Computational Statistics and Data Analysis, 71, 849–858.
54. J.Hannig, and T.C.M.Lee, (2009), “Generalized Fiducial Inference for Wavelet Regression,” Biometrika, 96, 847–860.
55. J.Hannig, C.M.Wang, and H.K.Iyer, (2003), “Uncertainty Calculation for the Ratio of Dependent Measurements,” Metrologia, 4, 177–186.
56. J.Hannig, and M.Xie, (2012), “A Note on Dempster-Shafer Recombinations of Confidence Distributions,” Electrical Journal of Statistics, 6, 1943–1966.
57. J.A.Hoeting, D.Madigan, A.E.Raftery, and C.T.Volinsky, (1999), “Bayesian Model Averaging:A Tutorial” (with discussion), Statistical Science, 14, 382–417; corrected version available at http://www.stat.washington.edu/www/research/online/hoetingl999.pdf.
58. H.K.Iyer, and P.Patterson, (2002), “A Recipe for Constructing Generalized Pivotal Quantities and Generalized Confidence Intervals,” Technical Report 10, Department of Statistics, Colorado State University, Fort Collins CO.
59. H.K.Iyer, C.M.J.Wang, and T.Mathew, (2004), “Models and Confidence Intervals for True Values in Interlaboratory Trials,” Journal of the American Statistical Association, 99, 1060–1071.
60. H.Jeffreys, (1940), “Note on the Behrens-Fisher Formula,” The Annals of Eugenics, 10, 48–51.
61. R.C.S.Lai, J.Hannig, and T.C.M.Lee, (2015), “Generalized Fiducial Inference for Ultra-High Dimensional Regression,” Journal of American Statistical Association, 110, 760–772.
62. E.Lawrence, C.Liu, Vander S.Wiel, and J.Zhang, (2009), “A New Method for Multinomial Inference Using Dempster-Shafer Theory.
63. T.C.M.Lee, (2002), “Tree-Based Wavelet Regression for Correlated Data using the Minimum Description Length Principle,” Australian and New Zealand Journal of Statistics, 44, 23–39.
64. D.V.Lindley, (1958), “Fiducial Distributions and Bayes’ Theorem,” Journal of the Royal Statistical Society, Series B, 20, 102–107.
65. Y.Liu, and J.Hannig, (2016), “Generalized Fiducial Inference for Binary Logistic Item Response Models,” Psychometrica, 81, 290–324.
66. P.A.Majumder, and J.Hannig, (2015), “Higher Order Asymptotics for Generalized Fiducial Inference,” unpublished manuscript.
67. R.Martin, and C.Liu, (2013), “Inferential Models:A Framework for Prior-Free Posterior Probabilistic Inference,” Journal of the American Statistical Association, 108, 301–313.
68. ——— (2015a), “Conditional Inferential Models:Combining Information for Prior-Free Probabilistic Inference,” Journal of the Royal Statistical Society, Series B, 77, 195–217.
69. ——— (2015b), Inferential Models:Reasoning with Uncertainty (Vol. 145), Boca Raton, FL:CRC Press.
70. ——— (2015c), “Marginal Inferential Models:Prior-Free Probabilistic Inference on Interest Parameters,” Journal of the American Statistical Association, 110, 1621–1631.
71. R.Martin, and S.G.Walker, (2014), “Asymptotically Minimax Empirical Bayes Estimation of a Sparse Normal Mean Vector,” Electronic Journal of Statistics, 8, 2188–2206.
72. R.Martin, J.Zhang, and C.Liu, (2010), “Dempster-Shafer Theory and Statistical Inference With Weak Beliefs,” Statistical Science, 25, 72–87.
73. R.J.McNally, H.K.Iyer, and T.Mathew, (2003), “Tests for Individual and Population Bioequivalence Based on Generalized p-Values,” Statistics in Medicine, 22, 31–53.
74. P.Patterson, J.Hannig, and H.K.Iyer, (2004), “Fiducial Generalized Confidence Intervals for Proportionof Conformance,” Technical Report 2004/11, Colorado State University, Fort Collins, CO.
75. D.Salome, (1998), “Statistical Inference via Fiducial Methods,” Ph.D. dissertation, University of Groningen, Groningen, The Netherlands.
76. T.Schweder, and N.L.Hjort, (2002), “Confidence and Likelihood,” Scandinavian Journal of Statistics, 29, 309–332.
77. K.Singh, M.Xie, and W.E.Strawderman, (2005), “Combining Information From Independent Sources Through Confidence Distributions,” The Annals of Statistics, 33, 159–183.
78. D.Sonderegger, and J.Hannig, (2014), “Fiducial Theory for Free-Knot Splines,” in Contemporary Developments in Statistical Theory, a Festschrift in honor of Professor Hira L. Koul, eds. S. Lahiri, A. Schick, A. SenGupta, T. N. Sriram, New York:Springer, pp. 155–189.
79. W.L.Stevens, (1950), “Fiducial Limits of the Parameter of a Discontinuous Distribution,” Biometrika, 37, 117–129.
80. G.Taraldsen, and B.H.Lindqvist, (2013), “Fiducial Theory and Optimal Inference,” The Annals of Statistics, 41, 323–341.
81. R.Tibshirani, (1996), “Regression Shrinkage and Selection via the Lasso,” Journal of the Royal Statistical Society, Series B, 58, 267–288.
82. K.-W.Tsui, and S.Weerahandi, (1989), “Generalized p-Values in Significance Testing of Hypotheses in the Presence of Nuisance Parameters,” Journal of the American Statistical Association, 84, 602–607.
83. ——— (1991), “Corrections:“Generalized p-Values in Significance Testing of Hypotheses in the Presence of Nuisance Parameters,”Journal of the American Statistical Association, 86, 256 (Journal of the American Statistical Association, 84, 1989, 602–607).
84. J.W.Tukey, (1957), “Some Examples With Fiducial Relevance,” The Annals of Mathematical Statistics, 28, 687–695.
85. P.Veronese, and E.Melilli, (2015), “Fiducial and Confidence Distributions for Real Exponential Families,” Scandinavian Journal of Statistics, 42, 471–484.
86. D.V.Wandler, and J.Hannig, (2011), “Fiducial Inference on the Maximum Mean of a Multivariate Normal Distribution,” Journal of Multivariate Analysis, 102, 87–104.
87. ——— (2012a), “A Fiducial Approach to Multiple Comparisons,” Journal of Statistical Planning and Inference, 142, 878–895.
88. ——— (2012b), “Generalized Fiducial Confidence Intervals for Extremes,” Extremes, 15, 67–87.
89. J.C.-M.Wang, J.Hannig, and H.K.Iyer, (2012a), “Fiducial Prediction Intervals,” Journal of Statistical Planning and Inference, 142, 1980–1990.
90. ——— (2012b), “Pivotal Methods in the Propagation of Distributions,” Metrologia, 49, 382–389.
91. J.C.-M.Wang, and H.K.Iyer, (2005), “Propagation of Uncertainties in Measurements Using Generalized Inference,” Metrologia, 42, 145–153.
92. ——— (2006a), “A Generalized Confidence Interval for a Measurand in the Presence of Type-A and Type-B Uncertainties,” Measurement, 39, 856–863.
93. ——— (2006b), “Uncertainty Analysis for Vector Measurands Using Fiducial Inference,” Metrologia, 43, 486–494.
94. Y.H.Wang, (2000), “Fiducial Intervals:What Are They?” The American Statistician, 54, 105–111.
95. S.Weerahandi, (1993), “Generalized Confidence Intervals,” Journal of the American Statistical Association, 88, 899–905.
96. ——— (1994), Correction:“Generalized Confidence Intervals,” Journal of the American Statistical Association, 89, 726(Journal of the American Statistical Association, 88, 1993, 899–905).
97. ——— (1995), Exact Statistical Methods for Data Analysis, Springer Series in Statistics, New York:Springer-Verlag.
98. B.L.Welch, and H.W.Peers, (1963), “On Formulae for Confidence Points Based on Integrals of Weighted Likelihoods,” Journal of the Royal Statistical Society, Series B, 25, 318–329.
99. G.N.Wilkinson, (1977), “On Resolving the Controversy in Statistical Inference,” Journal of the Royal Statistical Society, Series B, 39, 119–171.
100. M.Xie, R.Y.Liu, C.V.Damaraju, and W.H.Olson, (2013), “Incorporating External Information in Analyses of Clinical Trials With Binary Outcomes,” The Annals of Applied Statistics, 7, 342–368.
101. M.Xie, and K.Singh, (2013), “Confidence Distribution, the Frequentist Distribution Estimator of a Parameter:A Review,” International Statistical Review, 81, 3–39.
102. M.Xie, K.Singh, and W.E.Strawderman, (2011), “Confidence Distributions and a Unified Framework for Meta-Analysis,” Journal of the American Statistical Association, 106, 320–333.
103. X.Xu, and G.Li, (2006), “Fiducial Inference in the Pivotal Family of Distributions,” Science in China, Series A, 49, 410–432.
104. Yang, R., and Berger, J. O. (1997), “A Catalogue of Noninformative Priors,” Technical Report ISDS 97–42, Duke University, Durham, NC.
105. C.Zhang, and J.Huang, (2008), “The Sparsity and Bias of the Lasso Selection in High-Dimensional Linear Regression,” The Annals of Statistics, 36, 1567–1594.
106. J.Zhang, and C.Liu, (2011), “Dempster-Shafer Inference With Weak Beliefs,” Statistica Sinica, 21, 475–494.