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
Fisher introduced the the fiducial argument in 1930 [11]....
Controversy arose immediately. fisher had proposed the fiducial
argument as an alternative to the Bayesian argument of inverse
probability, which he condemned when no objective prior probability
could be stated.
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.
Later difficulties with the fiducial argument arose in cases of
multivariate estimation because of the nonuniqueness of the pivotals.
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,
In a 1963 conference on fiducial probability, Savage wrote 'The aim of
fiducial probability ... seems to be what I term making the Bayesian
omelet without breaking the Bayesian eggs.' In that sense, fiducial
probability is impossible. As with many great intellectual
contributions, what is of lasting value is what we learn trying to
understand Fisher's insights on fiducial probability. (See Edwards[4]
for much more on this theme.) His solution to the Behrens-Fisher
problem, for example, was a brilliant treatment of nuisance parameters
using Bayes' theorem. In this sense, "...the fiducial argument is
'learning from Fisher' [36, p.926]. Thus interpreted, it certainly
remains a valuable addition to staistical lore.
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
Following the 1930 publication, during the remaining 32 years of his
life, through two books and numerous articles , Fisher steadfastly
held to the idea captured in (1), and the reasoning leading to it
which we may call'fiducial inverse inference' then there is little
wonder that Fisher caused such puzzles with his novel idea
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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Society for Industrial and Applied Mathematics.
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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sample data. Ann. Math. Statist. 37, 355-374.
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unbalanced two-component normal mixed linear model. J. Amer. Statist. Assoc. 103, 854-
865.
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xxvi, 528-535.
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unknown parameters. Proceedings of the Royal Society of London A 139, 343-348.
Fisher, R. A. (1935a). The fiducial argument in statistical inference. Ann. Eugenics VI, 91-98.
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Sydney.
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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544 JAN HANNIG
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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
Best Answer
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).