Discussion of "Nonparametric generalized fiducial inference for survival functions under censoring"
aa r X i v : . [ s t a t . O T ] M a y Discussion of ‘Nonparametric generalized fiducialinference for survival functions under censoring’
G. Taraldsen and B.H. LindqvistDepartment of Mathematical SciencesNorwegian University of Science and TechnologyNTNU, NO-7491 Trondheim, [email protected] and [email protected] 27, 2019
Abstract
The following discussion is inspired by the paper
Nonparametric generalizedfiducial inference for survival functions under censoring by Cui and Hannig. Thediscussion consists of comments on the results, but also indicates it’s importancemore generally in the context of fiducial inference. A two page introduction tofiducial inference is given to provide a context.
Keywords:
Foundations and philosophical topics (62A01); Bayesian; Fiducial; Fre-quentist
We expect that many readers are not familiar with fiducial inference. This is in contrastto the well founded alternatives given by Bayesian and classical inference known to everystatistician today. Fiducial inference has not yet been established as a general theory, butthere has been considerable progress on this during the last decades, as also demonstratedby Cui and Hannig (2019). To discuss their contribution we need to provide a contextgiven by fiducial inference as we see it today.The original fiducial argument of Fisher (1930, p.532) starts by considering the relation u = F ( x ) (1)where F is the cumulative distribution function for the observation x . Fisher considersin particular the case where x is the empirical correlation of a sample of size n from the1ivariate Gaussian distribution. In this case F is strictly decreasing from 1 down to 0as a function of the unknown correlation θ . From this, Fisher argues that 1 − F ( x | θ )is the cumulative fiducial distribution for θ , and that π x ( θ ) = − ∂ θ F ( x | θ ) is the fiducialdensity of θ given x .Fisher’s argument uses the fact that equation (1) gives a correspondence between auniform law for u and the sampling law for x . The argument explains, in fact, thatthe percentiles of the fiducial distribution give confidence intervals, and hence that thefiducial distribution is a confidence distribution in this case. Even though Fisher himselfabandoned this interpretation in later works, it must be seen as one of the pioneeringworks that lead to the theory of confidence intervals and hypothesis testing as usedtoday. It is, as far as we know, the first paper that calculates exact confidence intervalsand explain them as such.Fiducial inference, in the version considered here, is given by replacing the relation (1)by a fiducial model x = θu (2)This economic notation is used by Dawid and Stone (1982, p.1055) when they define a functional model . It is a generalization of the structural models of Fraser (1968) whoconsiders the case where the model space Ω Θ is a group, and θu is the action of θ on u .Cui and Hannig (2019, eq.1) refer to equation (2) as a data generating equation . Samplesfrom a known distribution for u gives samples from the distribution of the observation x .In modern statistics, the possibility of simulating data from a statistical model is mostcentral, and any such algorithm is in fact a fiducial model.Equation (1) can be inverted to give x = θu = F − ( u ), where F depends on θ .Fisher’s initial model is hence a special case of a fiducial model. Consider for a momentthe following problem: The observation x is given and known to be generated from the fiducial model (2)by sampling u from a known distribution. How would You quantify Your un-certainty about the unknown model parameter θ ? It is clear that both u and θ are still uncertain, and it is reasonable, we claim, to quantifythese uncertainties by a joint distribution for ( u, θ ) such that equation (2) holds. Define θ = xu − to be a measurable selection solution of equation (2) for those ( x, u ) that allowsa solution. Assume, as we will exemplify below, that there exists a fiducial distributionfor u x derived from the original distribution of u and the observation x . A fiducialdistribution for the model θ can then be defined to be the distribution of θ x = x ( u x ) − (3)The fiducial distribution quantifies the uncertainty of θ given the assumed fiducial modeland given the observation x . This interpretation of the fiducial is what Fisher (1973,p.54-55) aimed at in his final writing on this: By contrast, the fiducial argument uses the observations only to change thelogical status of the parameter from one in which nothing is known of it, and no robability statement about it can be made, to the status of a random variablehaving a well-defined distribution. The correlation coefficient example treated initially by Fisher is such that the fiducialequation (2) defines a one-one correspondence between any two variables when the thirdis fixed. In this case, a simple fiducial model, the distribution of u x can be set equalto the original distribution of u . Fiducial samples are obtained simply by solving thefiducial equation for each sample u and returning the solution θ x = xu − .Another example is given by x = θu = θ + u , where θ is an element of a subspace Ω Θ of a Hilbert space Ω X . An important class of problems is obtained by letting Ω Θ be theimage space of the design matrix in linear regression. In this case, the fiducial equationwill fail to have solutions for all ( x, u ). Let P be the orthogonal projection on Ω Θ , andlet Q = 1 − P . Define the law of u x to be the conditional law of u given Qu = Qx . Thefiducial is then θ x = x [ u x ] − = x − u x .The previous example includes the general case of a location parameter, and in par-ticular inference based on sampling from the Gaussian distribution with unknown meanand known variance. As demonstrated by Fraser (1968), this can be seen as a particularcase of a group Ω Θ acting on the observation space Ω X , and cases with unknown variancecan also be included by considering other group actions. It follows in these cases, as alsofor the simple fiducial models, that the fiducial is a confidence distribution. Furthermore,Taraldsen and Lindqvist (2013) have proved that classical optimal actions, if they exist,are determined by the fiducial if the loss is invariant. Incidentally, the previous alsoexemplify a nonparametric fiducial in the sense given by an infinite dimensional Ω Θ .The previous indicate that a fiducial model (2) can be used to obtain a distributionwith interpretation similar to a Bayesian posterior as intended originally by Fisher. Italso show that confidence distributions and classical optimal actions can be obtained byfiducial arguments. Finally, a fiducial model (2) can also be used as a method for samplingfrom a Bayesian posterior. In a Bayesian set-up the joint distribution of ( u, θ ) is specified,and the distribution of u used above must be identified with the conditional distributionof u given θ . Sampling from the posterior can be done by sampling u conditionally given x and then θ given ( u, x ). In the case of group actions with prior equal to the rightinvariant prior this gives that the posterior coincides with the fiducial. Cui and Hannig (2019) consider failure distributions based on right censored data in anonparametric case. For simplicity, and since we will focus on the theoretical principles,we will focus on the uncensored case. Before leaving the censored case we will emphasizeits importance in applications, and add, as we see it, that the fiducial model for this caseis most natural. The ease of including this in the analysis is by itself a most convincingargument for the success of fiducial inference as demonstrated by Cui and Hannig (2019).The obvious choice, in retrospect, is to base nonparametric fiducial inference onFisher’s original fiducial relation in equation (1). The data is given by an ordered sam-ple x that obeys the fiducial relation u i = F ( x i ), or equivalently the fiducial model3 i = F − ( u i ). Here u ≤ · · · ≤ u n is the order statistic of a random sample from theuniform distribution on [0 , F is given by a measurable selection solution of this fiducial relation.We can and will restrict attention to the case where it is assumed that F is absolutelycontinuous in accordance with Cui and Hannig (2019, Assumption 2). In this case itfollows hence that the fiducial distribution for u x equals the original distribution for u asin Fishers original fiducial argument for the correlation coefficient. In contrast to Fishersoriginal argument there is here an infinity of possible randomized measurable selectionsolutions. It can, additionally, be observed that the given fiducial model is equivalentwith a group model x = θv : Ω Θ is the group of increasing and differentiable transfor-mations θ of the positive real line and v ≤ · · · ≤ v n is the order statistic of a randomsample from the standard exponential distribution.A particular absolutely continuous fiducial F I is determined by log-linear interpo-lation as described by Cui and Hannig (2019). This gives fiducial distributions for anyparameters of interest, and in particular for F ( x ) for a fixed x and the percentiles x α for a fixed α . The case with k samples can be treated similarly by the joint fiducial for F , . . . , F k . It is straightforward, in principle, to calculate corresponding fiducial intervalsor regions and corresponding fiducial p -values. This is exemplified by Cui and Hannig(2019) by a series of examples for k = 1 ,
2, and good frequentist properties are demon-strated as compared with existing methodology. The group model structure opens thequestion: Is optimal equivariant inference possible?The demonstrations, and the previous two paragraphs, constitute, in our opinion, themain message of the paper. Many more examples can, and should, be published based onconcrete applied problems, and the indicated natural route for nonparametric inference.An alternative approach is to take your favorite book on nonparametric inference andimplement and experiment with corresponding fiducial solutions.Proofs of stated coverage in the finite sample case are absent, but for k > k = 1 case seems possible to analyse completely, and the methodol-ogy should then be compared with similar results for the uncensored case presented bySchweder and Hjort (2016, Chap.11). It should be noted that Schweder and Hjort (2016)only consider confidence distributions for real valued parameters, and not for the unknown F itself. It is, in fact, unknown if the fiducial for F is a confidence distribution in a strictsense. The group model structure gives a starting point for investigating this further.All of these questions are related to the choice of a measurable selection solution. Isthere a natural choice? Is there a best choice? This question should be investigated inconcrete data situations. It can be observed that the choice F I is quick and convenient,but each realization is so special that it is not realistic in most situations. An alternative,which is still quick and convenient, is given by monotonic spline interpolation. Thefiducial distribution given by F I has defects when considered as a fiducial distribution for F , but the simulations demonstrate that resulting finite dimensional fiducials of certainfocus parameters have excellent properties.In summary, what is the possible role of the fiducial argument and distribution? The4ollowing Bayesian-Fiducial-Frequentist list give guidance: (B) Alternative algorithms for Bayesian analysis. (F)
A posterior fiducial state interpreted as Fisher intended. (F)
Alternative algorithms for frequentist analysis.All of this, seen in retrospect, is excellently presented and exemplified by Fraser (1968)for classical linear models. We believe that Cui and Hannig (2019) have taken the firstimportant step for similar results in the nonparametric case. Their main technical resultproves that the nonparametric fiducial is asymptotically a confidence distribution.
We take the opportunity of expressing our thanks for the invitation to comment on theinteresting and thought-provoking paper by Cui and Hannig (2019). This paper will serveas motivation for further developments of the theory of fiducial inference as initiated byFisher in the
Inverse probability paper from 1930. The importance of the 1930 paper byFisher, lies, according to Fisher (1950), in retrospect, in setting forth a new mode of rea-soning from observations to their hypothetical causes. We congratulate Cui and Hannigwith a successful demonstration of a fiducial argument in a nonparametric problem. Inconclusion, we can wholeheartedly and repeatedly agree with Efron (1998, p.107):
This is all quite speculative, but here is a safe prediction for the 21st century:statisticians will be asked to solve bigger and more complicated problems. Ibelieve that there is a good chance that objective Bayes methods will be de-veloped for such problems, and that something like fiducial inference will playan important role in this development. Maybe Fisher’s biggest blunder willbecome a big hit in the 21st century!
Additionally, we believe that the addition of nonparametric fiducial inference, as intro-duced by Cui and Hannig (2019), will play an important part of this adventure.
References
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The Annals of Statistics 10 (4), 1054–1074.Efron, B. (1998). R. A. Fisher in the 21st century (with discussion).
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Contributions to Mathematical Statistics . London: Chapman and Hall.5isher, R. A. (1930). Inverse probability.
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Statistical methods and scientific inference . Hafner press.Fraser, D. A. S. (1968).
The structure of inference . John Wiley.Schweder, T. and N. L. Hjort (2016).
Confidence, Likelihood, Probability: StatisticalInference with Confidence Distributions.
Cambridge University Press.Taraldsen, G. and B. H. Lindqvist (2013). Fiducial theory and optimal inference.