# International Encyclopedia of Statistical Science

2011 Edition
| Editors: Miodrag Lovric

# Fiducial Inference

• Jan Hannig
• Hari Iyer
• Thomas C. M. Lee
Reference work entry
DOI: https://doi.org/10.1007/978-3-642-04898-2_250

## Introduction

The origin of Generalized Fiducial Inference can be traced back to R. A. Fisher (Fisher 1930, 1933, 1935) who introduced the concept of a fiducial distribution for a parameter, and proposed the use of this fiducial distribution, in place of the Bayesian posterior distribution, for interval estimation of this parameter. In the case of a one-parameter family of distributions, Fisher gave the following definition for a fiducial density f( θ | x) of the parameter based on a single observation x for the case where the cdf F( x | θ) is a monotonic decreasing function of θ:
$$f(\theta /x) = -\frac{\partial F(x\vert \theta )} {\partial \theta } .$$
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## References and Further Reading

1. Barnard GA (1995) Pivotal models and the fiducial argument. Int Stat Rev 63:309–323
2. Berger JO, Sun D (2008) Objective priors for the bivariate normal model. Ann Stat 36:963–982
3. Casella G, Berger RL (2002) Statistical inference, 2nd edn. Wadsworth and Brooks/Cole, Pacific Grove, CAGoogle Scholar
4. Chiang A (2001) A simple general method for constructing confidence intervals for functions of variance components. Technometrics 43:356–367
5. Dawid AP, Stone M (1982) The functional-model basis of fiducial inference (with discussion). Ann Stat 10:1054–1074
6. Dawid AP, Stone M, Zidek JV (1973) Marginalization paradoxes in Bayesian and structural inference (with discussion). J R Stat Soc Ser B 35:189–233
7. Dempster AP (1966) New methods for reasoning towards posterior distributions based on sample data. Ann Math Stat 37:355–374
8. Dempster AP (1968) A generalization of Bayesian inference (with discussion). J R Stat Soc Ser B 30:205–247
9. Dempster AP (2008) The Dempster-Shafer calculus for statisticians. Int J Approx Reason 48:365–377
10. Fisher RA (1930) Inverse probability. Proc Cambridge Philos Soc 26:528–535
11. Fisher RA (1933) The concepts of inverse probability and fiducial probability referring to unknown parameters. Proc R Soc Lond A 139:343–348Google Scholar
12. Fisher RA (1935) The fiducial argument in statistical inference. Ann Eugenics 6:91–98Google Scholar
13. Fraser DAS (1961a) The fiducial method and invariance. Biometrika 48:261–280
14. Fraser DAS (1961b) On fiducial inference. Ann Math Stat 32:661–676
15. Fraser DAS (1966) Structural probability and a generalization. Biometrika 53:1–9
16. Fraser DAS (1968) The structure of inference. Wiley, New York
17. Glagovskiy YS (2006) Construction of fiducial confidence intervals for the mixture of cauchy and normal distributions. Master’s thesis, Department of Statistics, Colorado State UniversityGoogle Scholar
18. Grundy PM (1956) Fiducial distributions and prior distributions: an example in which the former cannot be associated with the latter. J R Stat Soc Ser B 18:217–221
19. Hannig J (2009a) On asymptotic properties of generalized fiducial inference for discretized data. Tech. Rep. UNC/STOR/09/02, Department of Statistics and Operations Research, The University of North CarolinaGoogle Scholar
20. Hannig J (2009b) On generalized fiducial inference. Stat Sinica 19:491–544
21. Hannig J, Abdel-Karim LEA, Iyer HK (2006a) Simultaneous fiducial generalized confidence intervals for ratios of means of lognormal distributions. Aust J Stat 35:261–269Google Scholar
22. Hannig J, Iyer HK, Patterson P (2006b) Fiducial generalized confidence intervals. J Am Stat Assoc 101:254–269
23. Hannig J, Iyer HK, Wang JC-M (2007) Fiducial approach to uncertainty assessment accounting for error due to instrument resolution. Metrologia 44:476–483Google Scholar
24. Hannig J, Lee TCM (2009) Generalized fiducial inference for wavelet regression. Biometrika 96(4):847–860
25. Hannig J, Wang CM, Iyer HK (2003) Uncertainty calculation for the ratio of dependent measurements. Metrologia, 4:177–186Google Scholar
26. Iyer HK, Patterson P (2002) A recipe for constructing generalized pivotal quantities and generalized confidence intervals. Tech. Rep. 2002/10, Department of Statistics, Colorado State UniversityGoogle Scholar
27. Iyer HK, Wang JC-M, Mathew T (2004) Models and confidence intervals for true values in interlaboratory trials. J Am Stat Assoc 99:1060–1071
28. Jeffreys H (1940) Note on the Behrens-Fisher formula. Ann Eugenics 10:48–51
29. Lidong E, Hannig J, Iyer HK (2008) Fiducial Intervals for variance components in an unbalanced two-component normal mixed linear model. J Am Stat Assoc 103:854–865
30. Lidong E, Hannig J, Iyer HK (2009) Fiducial generalized confidence interval for median lethal dose (LD50). (Preprint)Google Scholar
31. Lindley DV (1958) Fiducial distributions and Bayes’ theorem. J R Stat Soc Ser B 20:102–107
32. McNally RJ, Iyer HK, Mathew T (2003) Tests for individual and population bioequivalence based on generalized p-values. Stat Med 22:31–53Google Scholar
33. Patterson P, Hannig J, Iyer HK (2004) Fiducial generalized confidence intervals for proportion of conformance. Tech. Rep. 2004/11, Colorado State UniversityGoogle Scholar
34. Salome D (1998) Staristical inference via fiducial methods. Ph.D. thesis, University of GroningenGoogle Scholar
35. Stevens WL (1950) Fiducial limits of the parameter of a discontinuous distribution. Biometrika 37:117–129
36. Tsui K-W, Weerahandi S (1989) Generalized p-values in significance testing of hypotheses in the presence of nuisance parameters. J Am Stat Assoc 84:602–607
37. Tsui K-W, Weerahandi S (1991) Corrections: generalized p-values in significance testing of hypotheses in the presence of nuisance parameters. [J Am Stat Assoc 84 (1989), no. 406, 602–607; MR1010352 (90g:62047)]. J Am Stat Assoc 86:256Google Scholar
38. Wandler DV, Hannig J (2009) Fiducial inference on the maximum mean of a multivariate normal distribution (Preprint)Google Scholar
39. Wang JC-M, Iyer HK (2005) Propagation of uncertainties in measurements using generalized inference. Metrologia 42:145–153Google Scholar
40. Wang JC-M, Iyer HK (2006a) A generalized confidence interval for a measurand in the presence of type-A and type-B uncertainties. Measurement 39:856–863Google Scholar
41. Wang JC-M, Iyer HK (2006b) Uncertainty analysis for vector measurands using fiducial inference. Metrologia 43:486–494Google Scholar
42. Wang YH (2000) Fiducial intervals: what are they? Am Stat 54: 105–111Google Scholar
43. Weerahandi S (1993) Generalized confidence intervals. J Am Stat Assoc 88:899–905
44. Weerahandi S (1994) Correction: generalized confidence intervals [J Am Stat Assoc 88 (1993), no. 423, 899–905; MR1242940 (94e:62031)]. J Am Stat Assoc 89:726Google Scholar
45. Weerahandi S (1995) Exact statistical methods for data analysis. Springer series in statistics. Springer-Verlag. New YorkGoogle Scholar
46. Wilkinson GN (1977) On resolving the controversy in statistical inference (with discussion). J R Stat Soc Ser B 39:119–171
47. Xu X, Li G (2006) Fiducial inference in the pivotal family of distributions. Sci China Ser A Math 49:410–432

## Copyright information

© Springer-Verlag Berlin Heidelberg 2011

## Authors and Affiliations

• Jan Hannig
• 1
• Hari Iyer
• 2
• Thomas C. M. Lee
• 3
1. 1.The University of North Carolina at Chapel HillChapel HillUSA
2. 2.Colorado State UniversityFort CollinsUSA
3. 3.The University of California at DavisDavisUSA