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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?

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    $\begingroup$ Interesting question. Sprott (2000) says "Fiducial probability has not been widely accepted. This is mainly due to the fact that its unrestricted use produces contradictions. It is therefore important to underline the assumptions on which the above use of fiducial probability..." pp. 77. He also gives references about these contradictions such as Barnard (1987). This paper has been used to argue that Fisher "saw the light" and became a Bayesian. $\endgroup$
    – user10525
    Commented Apr 24, 2012 at 7:48
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    $\begingroup$ I thought I've read that Fisher never actually completed his fiducial argument or at least never made it quite consistent. An AMS article from 1964 by Dempster says that "It is concluded that the general form of the fiducial argument is appealing but that many of the restrictions imposed by Fisher are awkward or ambiguous and ought, perhaps, to be replaced." $\endgroup$
    – Wayne
    Commented May 3, 2012 at 2:43
  • $\begingroup$ @Wayne: The Dempster reference is eye opening. Thanks. $\endgroup$
    – JohnRos
    Commented May 3, 2012 at 8:51
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    $\begingroup$ When I was a graduate student at Stanford (giving away my age) some 35 years ago we had a seminar course "On Rereading Fisher." The title of the seminar came from a paper by that title that had been published a year or so earlier (maybe by Jimmie Savage). Anyway each student that was taking the seminar for a grade had to read one of Fisher's papers and report on it. Mine was on one about the famous Behrens-Fisher problem. The fiducial argument was prominent in that paper. My memory of the paper and the class is not sharp being that it was 35 years ago. $\endgroup$ Commented May 4, 2012 at 15:03
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    $\begingroup$ Fisher died in the 1960s in Australia. This was long before I became a statistician. I do think that Fisher thought fiducial theory was complete. I think other statisticians shot holes in it and he struggled to defend it. But if you have read Fisher you know that he was hard headed and always convinced that he was right (must of the time he was). I haven't seen Barnard's paper but I doubt that Fisher ever gave up on fiducial inference and I also doubt that he became a Bayesian. $\endgroup$ Commented May 4, 2012 at 15:04

5 Answers 5

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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.

Burch, B. D. and Iyer, H. K. (1997). Exact confidence intervals for a variance ratio (or heri- tability) in a mixed linear model. Biometrics 53, 1318-1333.

Burdick, R. K., Borror, C. M. and Montgomery, D. C. (2005a). Design and Analysis of Gauge R&R Studies. ASA-SIAM Series on Statistics and Applied Probability. Philadelphia, PA: 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.

Cai, T. T. (2005). One-sided confidence intervals in discrete distributions. J. Statist. Plann. Inference 131, 63-88.

Casella, G. and Berger, R. L. (2002). Statistical Inference. Wadsworth and Brooks/Cole Ad- vanced Books and Software, Pacific Grove, CA, second edn.

Daniels, L., Burdick, R. K. and Quiroz, J. (2005). Confidence Intervals in a Gauge R&R Study with Fixed Operators. J. Quality Tech. 37, 179-185.

Dawid, A. P. and Stone, M. (1982). The functional-model basis of fiducial inference. Ann. Statist. 10, 1054-1074. With discussions by G. A. Barnard and by D. A. S. Fraser, and a reply by the authors.

Dawid, A. P., Stone, M. and Zidek, J. V. (1973). Marginalization paradoxes in Bayesian and structural inference. J. Roy. Statist. Soc. Ser. B 35, 189-233. With discussion by D. J. Bartholomew, A. D. McLaren, D. V. Lindley, Bradley Efron, J. Dickey, G. N. Wilkinson, A. P.Dempster, D. V. Hinkley, M. R. Novick, Seymour Geisser, D. A. S. Fraser and A. Zellner, and a reply by A. P. Dawid, M. Stone, and J. V. Zidek.

Dempster, A. P. (1966). New methods for reasoning towards posterior distributions based on sample data. Ann. Math. Statist. 37, 355-374.

Dempster, A. P. (1968). A generalization of Bayesian inference. (With discussion). J. Roy. Statist. Soc. B 30, 205-247.

Dempster, A. P. (2008). The Dempster-Shafer calculus for statisticians. International Journal of Approximate Reasoning 48, 365-377.

E, L., Hannig, J. and Iyer, H. K. (2008). Fiducial intervals for variance components in an unbalanced two-component normal mixed linear model. J. Amer. Statist. Assoc. 103, 854- 865.

Efron, B. (1998). R. A. Fisher in the 21st century. Statist. Sci. 13, 95-122. With comments and a rejoinder by the author.

Fisher, R. A. (1930). Inverse probability. Proceedings of the Cambridge Philosophical Society xxvi, 528-535.

Fisher, R. A. (1933). The concepts of inverse probability and fiducial probability referring to 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.

Fisher, R. A. (1935b). The logic of inductive inference. J. Roy. Statist. Soc. B 98, 29-82.

Fraser, D. A. S. (1961). On fiducial inference. Ann. Math. Statist. 32, 661-676.

Fraser, D. A. S. (1966). Structural probability and a generalization. Biometrika 53, 1–9.

Fraser, D. A. S. (1968). The Structure of Inference. John Wiley & Sons, New York-London- 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

Ghosh, J. K. (1994). Higher Order Assymptotics. NSF-CBMS Regional Conference Series. Hay- ward: Institute of Mathematical Statistics.

Ghosh, J. K. and Ramamoorthi, R. V. (2003). Bayesian Nonparametrics. Springer Series in Statistics. Springer-Verlag, New York.

Glagovskiy, Y. S. (2006). Construction of Fiducial Confidence Intervals For the Mixture of Cauchy and Normal Distributions. Master’s thesis, Department of Statistics, Colorado State University.

Grundy, P. M. (1956). Fiducial distributions and prior distributions: an example in which the former cannot be associated with the latter. J. Roy. Statist. Soc. Ser. B 18, 217-221.

GUM (1995). Guide to the Expression of Uncertainty in Measurement. International Organiza- tion for Standardization (ISO), Geneva, Switzerland.

Hamada, M. and Weerahandi, S. (2000). Measurement system assessment via generalized infer- ence. J. Quality Tech. 32, 241-253.

Hannig, J. (1996). On conditional distributions as limits of martingales. Mgr. thesis, (in czech), Charles University, Prague, Czech Republic.

Hannig, J., E, L., Abdel-Karim, A. and Iyer, H. K. (2006a) Simultaneous fiducial generalized confidence intervals for ratios of means of lognormal distributions. Austral. J. Statist. 35, 261-269.

Hannig, J., Iyer, H. K. and Patterson, P. (2006b) Fiducial generalized confidence intervals. J. Amer. Statist. Assoc. 101, 254-269.

Hannig, J. and Lee, T. C. M. (2007). Generalized fiducial inference for wavelet regression. Tech. rep., Colorado State University.

Iyer, H. K. and Patterson, P. (2002). A recipe for constructing generalized pivotal quantities and generalized confidence intervals. Tech. Rep. 2002/10, Department of Statistics, Colorado State University.

Iyer, H. K., Wang, C. M. J. and Mathew, T. (2004). Models and confidence intervals for true values in interlaboratory trials. J. Amer. Statist. Assoc. 99, 1060-1071.

Jeffreys, H. (1940). Note on the Behrens-Fisher formula. Ann. Eugenics 10, 48-51.

Jeffreys, H. (1961). Theory of Probability. Clarendon Press, Oxford, third edn.

Le Cam, L. and Yang, G. L. (2000). Asymptotics in Statistics. Springer Series in Statistics. New York: Springer-Verlag, second edn.

Liao, C. T. and Iyer, H. K. (2004). A tolerance interval for the normal distribution with several variance components. Statist. Sinica 14, 217-229.

Lindley, D. V. (1958). Fiducial distributions and Bayes’ theorem. J. Roy. Statist. Soc. Ser. B 20, 102-107.

McNally, R. J., Iyer, H. K. and Mathew, T. (2003). Tests for individual and population bioe- quivalence based on generalized p-values. Statistics in Medicine 22, 31-53.

Mood, A. M., Graybill, F. A. and Boes, D. C. (1974). Introduction to the Theory of Statistics. McGraw-Hill, third edn.

Pounds, S. and Morris, S. W. (2003). Estimating the occurrence of false positives and false neg- atives in microarray studies by approximating and partitioning the empirical distribution of p-values. Bioinformatics 19, 123601242.

Salome, D. (1998). Staristical Inference via Fiducial Methods. Ph.D. thesis, University of Gronin- gen. 544 JAN HANNIG

Searle, S. R., Casella, G. and McCulloch, C. E. (1992). Variance Components. John Wiley & Sons, New York.

Stevens, W. L. (1950). Fiducial limits of the parameter of a discontinuous distribution. Biometrika 37, 117-129.

Tsui, K.-W. and Weerahandi, S. (1989). Generalized p-values in significance testing of hypothe- ses in the presence of nuisance parameters. J. Amer. Statist. Assoc. 84, 602-607.

Wang, C. M. and Iyer, H. K. (2005). Propagation of uncertainties in measurements using gen- eralized inference. Metrologia 42, 145-153.

Wang, C. M. and Iyer, H. K. (2006a). A generalized confidence interval for a measurand in the presence of type-A and type-B uncertainties. Measurement 39, 856–863. Wang, C. M. and Iyer, H. K. (2006b). Uncertainty analysis for vector measurands using fiducial inference. Metrologia 43, 486-494.

Weerahandi, S. (1993). Generalized confidence intervals. J. Amer. Statist. Assoc. 88, 899-905.

Weerahandi, S. (2004). Generalized Inference in Repeated Measures. Wiley, Hoboken, NJ.

Wilkinson, G. N. (1977). On resolving the controversy in statistical inference. J. Roy. Statist. Soc. Ser. B 39, 119-171. With discussion.

Yeo, I.-K. and Johnson, R. A. (2001). A uniform strong law of large numbers for U-statistics with application to transforming to near symmetry. Statist. Probab. Lett. 51, 63-69.

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: [email protected] (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

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    $\begingroup$ @MichaelChernick: I was hoping for an explanation of the argument and it's flaws. I do not feel current answers, while very useful, are complete. $\endgroup$
    – JohnRos
    Commented May 7, 2012 at 19:39
  • $\begingroup$ Some of that is described in my excerpt from the Encyclopedia of Biostatistics and certainly just about anything you want to know is in the slew of references that I have cited. I bring up Hannig's work because I discovered it by surprise while looking at what would come up from a Google search. I haven't read his stuff yet but I think it is different from other things which either describe the method and the controversy or point to the flaws. Hannig is doing serious research at modifying fiducial inference to make it work. $\endgroup$ Commented May 7, 2012 at 19:50
  • $\begingroup$ I think that is very interesting whether or not it is what you were looking for. I also think that questions like the one you posed can easily be answered by Google searches. The encyclopedia article explains the fiducial argument in great detail. You can go back to Fisher's 1930 paper to see how he originally stated it. Also it is probably discussed in Joan Fisher Box's biography of Fisher. What more could you possibly want? $\endgroup$ Commented May 7, 2012 at 19:59
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    $\begingroup$ @JohnRos: I added to my answer which I think makes mine complete. In general i feel that giving someone a specific reference that provides a complete answer is complete enough. I think that the poser of the question who is really interested in the answwer should go to the trouble of looking at the references and learn that way. We are all grown ups and we don't have to be spoon fed everything. $\endgroup$ Commented May 7, 2012 at 21:55
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Fiducial inference sometimes interprets likelihoods as probabilities for the parameter $\theta$. That is, $M(x)L(\theta|x)$, provided that $M(x)$ is finite, is interpreted as a probability density function for $\theta$ in which $L(\theta|x)$ is the likelihood function of $\theta$ and $M(x)=(\int_{-\infty}^{\infty}L(\theta|x)d\theta)^{-1}$. You can see Casella and Berger, pages 291-2, for more details.

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  • $\begingroup$ Casella and Berger make this statement, but it is not entirely correct to consider the likelihood function as equal to the fiducial distribution. This is only the case for estimation of scale parameters or location parameters. For the estimation of a rate parameter it becomes false. See also stats.stackexchange.com/a/592453/164061 $\endgroup$ Commented Oct 19, 2022 at 7:53
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Just to add to what is said, there was controversy between Fisher and Neyman about significance testing and interval estimation. Neyman defined confidence intervals while Fisher introduced fiducial intervals. They argued differently about their construction but the constructed intervals were usually the same. So the difference in the definitions was largely ignored until it was discovered that they differed when dealing with the Behrens-Fisher problem. Fisher argued adamantly for the fiducial approach but inspite of his brillance and his strong advocation of the method, there appeared to be flaws and since the statistical community considers it discredited it is not commonly discussed or used. The Bayesian and frequentist approaches to inference are the two that remain.

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TL;DR

The fiducial argument has not been accepted because the idea doesn't work.

The fiducial distribution is disguised as something that looks like a probability distribution (and people might have wanted it to behave like a probability distribution) but it is not the same as a probability distribution. It is only a function of probabilities.

You can not do the same thing with a fiducial distribution as with, for instance, a posterior probability density. This is illustrated in the example below where we compute a 80% fiducial interval but get in some cases a 100% coverage.

Example where it doesn't work

In the nice answer by Michael R. Chernick it is mentioned that the logic behind the fiducial distribution started to fail when people tried to apply it in a multidimensional setting like for instance the Behrens Fisher problem. Here we give a one dimensional example that already shows that it does not work.

Let some statistic $X$ be distributed as a Uniform distribution with

$$X \sim \mathcal{U}\left(\theta-0.5\sqrt{1+\theta^2},\theta+0.5\sqrt{1+\theta^2}\right)$$

We can plot the cumulative distribution function (CDF) $F(x;\theta)$ as a function of $x$ and $\theta$ using isolines.

cdf as isolines with 80% fiducial interval boundaries

The image shows the CDF as function of $x$ and $\theta$

  • In vertical direction, for fixed $\theta$, the function describes the CDF of the observation $x$ which is a random variable.

    We can compute the probability density function as $$\frac{\partial}{\partial x} F(x;\theta)$$

  • In horizontal direction, for fixed $x$, the function describes the fiducial distribution for the estimation of $\theta$

    We can compute the fiducial density as $$\frac{\partial}{\partial\theta} F(x;\theta)$$

For example the fiducial density, when we observed $x=0.1$ looks like.

fiducial density

The points in red are inside the 80% interval [-0.396 , 0.475], it is the 80% probability mass with the highest density.

The problem is the following: the probability statements by the fiducial distribution only 'work' when we consider the same quantiles, independent from the observation $x$. However when change the quantiles as function of $x$ then the content of the probability statements entailed by the fiducial distribution are changed and become false. Therefore, the distribution can not be used in a similar way as a probability density. An example when this happens is for instance like the example above when we compute a highest density interval in combination in combination with a fiducial density that doesn't have the same shape for different observations $x$ (which makes us select different quantiles).

We can see this in the plot when $\theta = 0$ for that case we get a 100% coverage by the 80% highest density interval instead of a 80% coverage. This is not what you would expect when the fiducial distribution could be used as a probability density for $\theta$.

R-code for the two plots:

#### parameters for drawing
d = 0.01
t = seq(-2,2.6,d)
t2 = seq(3.4,4,d)
tm = 3
grey = rgb(0.3,0.3,0.3)

### empty canvas
plot(-10,-10, xlim = c(-2,4), ylim = c(-3,5),
     xlab = expression(theta), ylab = "x", main = "example of highest density 80% feducial interval")

### add isolines 
for (q in c(0:10)*0.1) {
   lines(t,t+(q-0.5)*sqrt(1+t^2), col = grey)
   lines(t2,t2+(q-0.5)*sqrt(1+t2^2), col = grey)
   text(tm,tm+(q-0.5)*sqrt(1+tm^2),
        bquote(F(x*";"*theta)==.(q)), col = grey, cex = 0.6, srt = 15+35*q)
}

fiducial = function(x, plotting = TRUE, alpha = 0.8) {
  dt = 0.001
  ### domain of fiducial distribution 
  tmin = (1/3) * (4*x - sqrt(4*x^2 + 3))
  tmax = (1/3) * (4*x + sqrt(4*x^2 + 3))
  ts = seq(tmin, tmax, dt)

  ### compute distribution
  f = (ts*x + 1)/(ts^2+1)^1.5

  ### calculate highest density region by ordering densities
  ord = order(f)       #order
  p = cumsum(f[ord])*dt   #cumulative probability 
  sel = which(p<1-alpha) # select complement of highest alpha% density
  output = range(ts[ord][-sel]) ## range of the highest alpha% density interval

  ### example plot of density
  if(plotting == TRUE) {
     plot(ts,f, col = 1 + (ts >= output[1]) * (ts <= output[2]))
  }
  output
}

### compute intervals as function of observed x

xs = seq(-4,5,0.01)
low = c()    ### empty array that will be filled
high = c()
for (x in xs) {
   interval = fiducial(x, plotting = FALSE)
   low = c(low, interval[1])
   high = c(high, interval[2])
}

### add curves for fiducial interval
lines(low[low<2.6],xs[low<2.6], lwd =2)
lines(high[high<2.6],xs[high<2.6], lwd =2)

### add example interval for observation X = 0.1
int = fiducial(0.1, plotting = 0)
#lines(c(-2,2),c(0.1,0.1), lty = 2)
lines(int,c(0.1,0.1), lwd =2, col = 2)
points(int, c(0.1,0.1), pch = 21, col = 2, bg = 0, cex = 0.7)
text(0.8,0.1,"example interval if x=0.1", pos = 4, col = 2)

### example plot
fiducial(0.1)
title("example fiducial distribution if x = 0.1 \n highest 80% density is highlighted in red")
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In a large undergraduate class of engineering intro stats at Georgia Tech, when discussing confidence intervals for the population mean with variance known, one student asked me (in the language of MATLAB): "Can I calculate the interval as > norminv([alpha/2,1-alpha/2], barX, sigma/sqrt(n))?" In translation: could he take $\frac{\alpha}{2}$ and $1-\frac{\alpha}{2}$ quantiles of a normal distribution centered at $\bar X$ with scale $\frac\sigma{\sqrt{n}}$?

I said – of course YES, pleasantly surprised that he naturally arrived to the concept fiducial distribution.

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  • $\begingroup$ This is not quite an answer... $\endgroup$ Commented Jul 15, 2020 at 3:17

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