Fiducial inference: Difference between revisions

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Fisher designed the fiducial method to meet perceived problems with the Bayesian approach, at a time when the frequentist approach had yet to be fully developed. Such problems related to the need to assign a [[prior distribution]] to the unknown values. The aim was to have a procedure, like the Bayesian method, whose results could still be given an inverse probability interpretation based on the actual data observed. The method proceeds by attempting to derive a "fiducial distribution", which is a measure of the degree of faith that can be put on any given value of the unknown parameter and is faithful to the data in the sense that the method uses all available information.
Unfortunately Fisher did not give a general definition of the fiducial method and he denied that the method could always be applied.{{Citation needed|date=June 2011}} His only examples were for a single parameter; different generalisations have been given when there are several parameters. A relatively complete presentation of the fiducial approach to inference is given by Quenouille (1958), while Williams (1959) describes the application of fiducial analysis to the [[Calibration (statistics)|calibration]] problem (also known as "inverse regression") in [[regression analysis]].<ref>Williams (1959, Chapter 6)</ref> Further discussion of fiducial inference is given by Kendall & Stuart (1973).<ref name=KS>Kendall, M.G., Stuart, A. (1973) ''The Advanced Theory of Statistics, Volume 2: Inference and Relationship, 3rd Edition'', Griffin. {{ISBN |0-85264-215-6}} (Chapter 21)</ref>
==The fiducial distribution==
*[[David R. Cox|Cox, D. R.]] (2006). ''Principles of Statistical Inference'', CUP. {{ISBN |0-521-68567-2}}.
*{{cite book |last=Fisher |first=R A |coauthors= |title=Statistical Methods and Scientific Inference|year=1956 |publisher=Hafner |location=New York |isbn=0-02-844740-9}}
* [[Ronald A. Fisher|Fisher, Ronald]] "Statistical methods and scientific induction" ''Journal of the Royal Statistical Society, Series B'' 17 (1955), 69—78. (criticism of statistical theories of [[Jerzy Neyman]] and [[Abraham Wald]] from a fiducial perspective)
*Quenouille, M.H. (1958) ''Fundamentals of Statistical Reasoning''. Griffin, London
*Williams, E.J. (1959) ''Regression Analysis'', Wiley {{LCCN|59011815}}
*Young, G.A., Smith, R.L. (2005) ''Essentials of Statistical Inference'', CUP. {{ISBN |0-521-83971-8}}
* Fraser, D.A.S. (1961) "The fiducial method and invariance." ''Biometrika'', '''48''', 261-80.
* Fraser, D.A.S. (1961) "On fiducial inference." ''Annals of Mathematical Statistics,'' '''32''', 661-676.