One of the late contributions of R.A. Fisher was fiducial intervals and fiducial principled arguments. This approach however is nowhere near as popular as frequentist or Bayesian principled arguments.
What is the fiducial argument and why has is not been accepted?
Solved – the fiducial argument and why has it not been accepted
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I may have mentioned this on the site once before. I will try to find a link to a post where I discussed this. Around 1977 when I was a graduate student at Stanford we had a Fisher seminar that I enrolled in. A number of Stanford professors and visitors participated including Brad Efron and visitors Seymour Geisser and David Hinkley. Jimmie Savage had just at that time published an article with the title "On Rereading R. A. Fisher" in Annal of Statistics I think. Since you are so interested in Fisher I recommend you find and read this paper.
Motivated by the paper the seminar was designed to reread many of Fisher's famous papers. My assignment was the article on the Behrens-Fisher problem. My feeling is that Fisher was vain and stubborn but never foolish. He had great geometric intuition and at times had difficulty communicating with others. He had a very cordial relationship with Gosset but harsh disagreements with Karl Pearson (maximum likelihood vs method of moments) and with Neyman and Egon Pearson (significance testing via fiducial inference vs the Neyman-Pearson approach to hypothesis testing). Although the fiducial argument is generally considered to be Fisher's only big flaw and has been discredited, the approach is not totally dead and there has been new research in it in recent years.
I think that fiducial inference was Fisher's way to try to be an "objective Bayesian". I am sure he thought long and hard about the statistical foundations. He didn't accept the Bayesian approach but also did not see the idea of basing inference on considering the possible samples that you didn't draw as making sense either. He believed that inference should be based only on the data at hand. This idea is a lot like Bayesian inference in that the Bayesians draw inference based soley on the data (the likelihood) and the parameters (the prior distribution). Fisher in my view was thinking a lot like Jeffreys except that he wanted inference to be based on the likelihood and wanted to dispense with the prior altogether. That is what led to fiducial inference.
The Biography by Fisher's daughter Joan Fisher Box
R A Fisher An Appreciation, Hinkley and Feinberg editors
A book by Erich Lehmann about Fisher and Neyman and the birth of Classical Statistics
This is a link to an earlier post that I commented on that you also posted. Behrens–Fisher problem
In conclusion I think I need to address your short question. If the statement you quoted "Fisher approximated the distribution of this by ignoring the random variation of the relative sizes of the standard deviations" is what you are referring to I think that is totally false. Fisher never ignored variation. I reiterate that I think the fiducial argument was grounded in the idea that the observed data and the likelihood function should be the basis of inference and not the other samples that you could have gotten from the population distribution. So I would side with you on this one. With respect to Bartlett as I recall from my study of this so many years ago, they also had heated debates on this and Bartlett made a good case and held his own in the debate.
The fiducial argument is to interpret likelihood as a probability. Even if likelihood measures the plausibility of an event, it does not satisfy the axioms of probability measures (in particular there is no guarantee that it sums to 1), which is one of the reasons this concept was never so successful.
Let's give an example. Imagine that you want to estimate a parameter, say the half-life $\lambda$ of a radioactive element. You take a couple of measurements, say $(x_1, \ldots, x_n)$ from which you try to infer the value of $\lambda$. In the view of the traditional or frequentist approach, $\lambda$ is not a random quantity. It is an unknown constant with likelihood function $\lambda^n \prod_{i=1}^n e^{-\lambda x_i} = \lambda^n e^{-\lambda(x_1+\ldots+x_n)}$.
In the view of the Bayesian approach, $\lambda$ is a random variable with a prior distribution; the measurements $(x_1, \ldots, x_n)$ are needed to deduce the posterior distribution. For instance, if my prior belief about the value of lambda is well represented by the density distribution $2.3 \cdot e^{-2.3\lambda}$, the joint distribution is the product of the two, i.e. $2.3 \cdot \lambda^n e^{-\lambda(2.3+x_1+\ldots+x_n) }$. The posterior is the distribution of $\lambda$ given the measurements, which is computed with Bayes formula. In this case, $\lambda$ has a Gamma distribution with parameters $n$ and $2.3+x_1+\ldots+x_n$.
In the view of fiducial inference, $\lambda$ is also a random variable but it does not have a prior distribution, just a fiducial distribution that depends only on $(x_1, \ldots, x_n)$. To follow up on the example above, the fiducial distribution is $\lambda^n e^{-\lambda(x_1+\ldots+x_n)}$. This is the same as the likelihood, except that it is now interpreted as a probability. With proper scaling, it is a Gamma distribution with parameters $n$ and $x_1+\ldots+x_n$.
Those differences have most noticeable effects in the context of confidence interval estimation. A 95% confidence interval in the classical sense is a construction that has 95% chance of containing the target value before any data is collected. However, for a fiducial statistician, a 95% confidence interval is a set that has 95% chance of containing the target value (which is a typical misinterpretation of the students of the frequentist approach).
Best Answer
I am surprised that you don't consider us authorities. Here is a good reference: Encyclopedia of Biostatistics, Volume 2, page 1526; article titled "Fisher, Ronald Aylmer." Starting at the bottom of the first column on the page and going through most of the second column the authors Joan Fisher Box (R. A. Fisher's daughter) and A. W. F. Edwards write
They go on to discuss the debates with Jeffreys and Neyman (particularly Neyman on confidence intervals). The Neyman-Pearson theory of hypothesis testing and confidence intervals came out in the 1930s after Fisher's article. A key sentence followed.
In the same volume of the Encyclopedia of Biostatistics there is an article pp. 1510-1515 titled "Fiducial Probability" by Teddy Seidenfeld which covers the method in detail and compares fiducial intervals to confidence intervals. To quote from the last paragraph of that article,
I think in these last few sentences Edwards is trying to put a favorable light on Fisher even though his theory was discredited. I am sure that you can find a wealth of information on this by going through these encyclopedia papers and similar ones in other statistics papers as well as biographical articles and books on Fisher.
Some other references
Box, J. Fisher (1978). "T. A. Fisher: The Life of a Scientist." Wiley, New York Fisher, R. A. (1930) Inverse Probability. Proceedings of the Cambridge Philosophical Society. 26, 528-535.
Bennett, J. H. editor (1990) Statistical Inference and Analysis: Selected Correspondence of R. A. Fisher. Clarendon Press, Oxford.
Edwards, A. W. F. (1995). Fiducial inference and the fundamental theorm of natural selection. Biometrics 51,799-809.
Savage L. J. (1963) Discussion. Bulletin of the International Statistical Institute 40, 925-927.
Seidenfeld, T. (1979). "Philosophical Problems of Statistical Inference" Reidel, Dordrecht . Seidenfeld, T. (1992). R. A. Fisher's fiducial argument and Bayes' theorem. Statistical Science 7, 358-368.
Tukey, J. W. (1957). Some examples with fiducial relevance. Annals of Mathematical Statistics 28, 687-695.
Zabell, S. L. (1992). R. A. Fisher and the fiducial argument. Statistical Science 7, 369-387.
The cocept is difficult to understand because fisher kept changing it as Seidenfeld said in his article in the Encyclopedia of Biostatistics
Equation (1) that Seidenfeld refers to is the fiducial distribution of the parameter $\theta$ given $x$ as $\text{fid}(\theta|x) \propto \partial F/\partial \theta$ where $F(x,\theta)$ denotes a one-parameter cumulative distribution function for the random variable $X$ at $x$ with parameter $\theta$. At least this was Fisher's initial definition. Later it got extended to multiple parameters and that is where the trouble began with the nuisance parameter $\sigma$ in the Behrens-Fisher problem. So a fiducial distribution is like a posterior distribution for the parameter $\theta$ given the observed data $x$. But it is constructed without the inclusion of a prior distribution on $\theta$.
I went to some trouble getting all this but it is not hard to find. We are really not needed to answer questions like this. A Google search with key words "fiducial inference" would likely show everything I found and a whole lot more.
I did a Google search and found that a UNC Professor Jan Hannig has generalized fiducial inference in an attempt to improve it. A Google search yields a number of his recent papers and a powerpoint presentation. I am going to copy and paste the last two slides from his presentation below:
Concluding Remarks
Generalized fiducial distributions lead often to attractive solution with asymptotically correct frequentist coverage.
Many simulation studies show that generalized fiducial solutions have very good small sample properties.
Current popularity of generalized inference in some applied circles suggests that if computers were available 70 years ago, fiducial inference might not have been rejected.
Quotes
Zabell (1992) “Fiducial inference stands as R. A. Fisher’s one great failure.” Efron (1998) “Maybe Fisher’s biggest blunder will become a big hit in the 21st century! "
Just to add more references, here is the reference list I have taken from Hannig's 2009 Statistics Sinica paper. Pardon the repetition but I think this will be helpful.
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Burdick, R. K., Park, Y.-J., Montgomery, D. C. and Borror, C. M. (2005b). Confidence intervals for misclassification rates in a gauge R&R study. J. Quality Tech. 37, 294-303.
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Fraser, D. A. S. (2006). Fiducial inference. In The New Palgrave Dictionary of Economics (Edited by S. Durlauf and L. Blume). Palgrave Macmillan, 2nd edition. ON GENERALIZED FIDUCIAL INFERENCE 543
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Zabell, S. L. (1992). R. A. Fisher and the fiducial argument. Statist. Sci. 7, 369-387. Department of Statistics and Operations Research, The University of North Carolina at Chapel Hill, Chapel Hill, NC 27599-3260, U.S.A. E-mail: hannig@unc.edu (Received November 2006; accepted December 2007)
The article i got this from is Statistica Sinica 19 (2009), 491-544 ON GENERALIZED FIDUCIAL INFERENCE∗ Jan Hannig The University of North Carolina at Chapel Hill