1 Introduction
A Bayesian posterior is said to be a fiducial posterior if it coincides with a fiducial distribution. The question of existence of a Bayesian prior such that the resulting posterior is a fiducial posterior has attracted interest since the introduction of the fiducial argument by Fisher (1930, 1935). Cases where the fiducial is not a Bayesian posterior are interesting because the fiducial theory then brings truly new armory for the construction of new inference procedures. The cases where there is a fiducial posterior are interesting because the corresponding fiducial algorithm can be simpler to implement than the competitors based on the Bayesian theory.
The best result in the onedimensional case was found by Lindley (1958). He proved that, given appropriate smoothness conditions, a fiducial posterior exists if and only if the problem can be transformed by oneone transformations of the parameter and sample space into the standard location problem. The best result obtained so far in the multivariate case was found by Fraser (1961b, a). For a group model where a right Haar measure exists, the fiducial coincides with the posterior from the right Haar measure as a Bayesian prior. The main result in this paper is Theorem 1 that contains both results as special cases.
2 Fiducial posteriors
The arguments in the following make it necessary to include improper priors in the considerations, and this will here be done based on the theory presented by Taraldsen and Lindqvist (2010). A brief summary of the necessary ingredients from this theory is given next.
Definition 1 (The basic space).
The basic space is equipped with a finite measure defined on the field of events.
All definitions and results in the following will implicitly or explicitly rely on the existence of the underlying basic space. This is as in the theory of probability presented by Kolmogorov (1933), but the requirement is here replaced by the weaker requirement that is finite: There exist events with and . The above can be summarized by saying that it is assumed throughout in this paper that the basic space is a finite measure space .
Definition 2 (Random element).
A random element in a measurable space is given by a measurable function . The law of is defined by
(1) 
The random element is finite if the law is finite.
Definition 2 corresponds to the definition by Lamperti (1966, p.4,eq.2) and the definition by Schervish (1995, p.607). It also corresponds to the original definition given by Kolmogorov (1933, p.21, eq.1), but he used superscript notation instead of the above subscript notation. The law will also be referred to as the distribution of . The term random quantity is used by Schervish (1995) and can be used as an alternative to the term random element used above and by Fréchet (1948). The term random variable is reserved for the case of a random real number. This is given by a measurable , where is the field generated by the open intervals.
Definition 2 of a random element is more general than any of the above given references since is not required to be a probability space, but it is assumed to be a finite measure space. The space
comes, however, equipped with a large family of conditional distributions that are true probability distributions. This is exactly what is needed for the formulation of a statistical inference problem, and will be explained next.
Let and be random elements, and assume that is finite. Existence of the conditional expectation and the factorization
(2) 
can then be established. The proof follows from the RadonNikodym theorem exactly as in the case where the underlying space is a probability space (Taraldsen and Lindqvist, 2010). The case gives in particular as a family of probability spaces. This last claim is not strictly true, but given appropriate regularity conditions there will exist a regular conditional law as claimed (Schervish, 1995, p.618).
A statistical model is usually defined to be a family of probability spaces. This definition is also used here, but with an added assumption included in the definition: It is assumed that there exist a random element , and a finite random element so that . It is in particular assumed that both the sample space and the model parameter space are measurable spaces. The law is not assumed to be known and is not specified. Similarly, the functions and are assumed to exist, but they are also not specified. This is by necessity since the underlying space is not specified. It is an abstract underlying space that makes it possible to formulate a consistent theory.
A Bayesian model is given by a statistical model and the additional specification of the law of . This prior law can be improper in the theory as just described, and discussed in more detail by Taraldsen and Lindqvist (2010). The posterior law is well defined if is
finite. The result of Bayesian inference is given by the posterior law, and Bayesian inference is hence trivial except for the practical difficulties involved in the calculation of the posterior and derived statistics. The most difficult part from a theoretical perspective is to justify the choice of statistical model and the prior in concrete modeling cases.
Fiducial arguments were invented by Fisher (1930, 1935) to tackle cases without a prior law, but with the aim to obtain a result similar to the posterior distribution. The resulting distribution from the fiducial argument is called a fiducial distribution. The following definition (Taraldsen and Lindqvist, 2013) will be used here. It should be noted that the definition uses concepts that rely on existence of the underlying basic space .
Definition 3 (Fiducial model and distribution).
Let be a finite random element in the model parameter space . A fiducial model is defined by a random element in the Monte Carlo space and a measurable function where is the sample space. The model is conventional if the conditional law does not depend on . The model is simple if the fiducial equation has a unique solution for all . If the model is both conventional and simple, then the fiducial distribution corresponding to an observation is the distribution of where .
A fiducial model is a fiducial model for the statistical model if
(3) 
The fiducial model gives a method for simulation from the statistical model: If is a sample from the known Monte Carlo law , then is a sample from . Sampling from the fiducial follows likewise, but involves solving the fiducial equation to obtain the sample . This, and related definitions in the literature, are discussed in more detail by Taraldsen and Lindqvist (2013).
We have now presented the necessary ingredients for the formulation of the main theoretical results here. The first result gives conditions that ensure that the fiducial coincides with a Bayesian posterior.
Theorem 1.
Assume that is a conventional simple fiducial model for the statistical model . If the Bayesian prior implies that distribution of does not depend on , then the Bayesian posterior distribution is well defined and identical with the fiducial distribution of .
It should in particular be observed that the required finiteness of is a part of the conclusion in the previous theorem. This ensures that the Bayesian posterior exists.
The next result gives a recipe for posterior sampling based on a fiducial model.
Theorem 2.
Assume that is a conventional fiducial model and that is finite for a given prior . Assume furthermore that for some finite measure and jointly measurable . If is a sample from a probability distribution proportional to and is a sample from the conditional law , then is a sample from the Bayesian posterior distribution of given .
3 The location problem
Assume that is the observed realization of a random variable where
(4) 
where is a sample from the conditional law . It is assumed that is known and does not depend on . The pair is then a fiducial model for the statistical model . The problem is to make statistical inference regarding the model parameter based on the model and the observation .
Consider first fiducial inference. The fiducial distribution is determined by the solution of the fiducial equation . Monte Carlo sampling from the known law gives corresponding samples
from the fiducial distribution. The mean and standard deviation can then be calculated with a precision depending on the choice of Monte Carlo size and the random number generator. This can then be reported as an estimate of
and a standard error respectively. A more complete report can be given by a direct Monte Carlo estimate of the fiducial distribution itself in the form of a graph. This represents then the state of knowledge regarding
based on the observation and the fiducial model.Consider next Bayesian inference. Assume for simplicity that , where is Lebesgue measure on the real line. If is the prior law, then the posterior law is given by a density where is a normalization constant. This normalization is generally possible if is finite, and this happens exactly when for (almost) all (Taraldsen and Lindqvist, 2010). It is always possible if is a probability density, but an alternative sufficient condition is that is bounded. The particular case gives the result . A simple calculus exercise shows directly that this coincides with the fiducial law derived above. The reporting of the result can be done as in the case of fiducial inference.
The previous result can also be inferred from Theorem 1 since the law of is the Lebesgue measure when the law of is the Lebesgue measure. More generally the assumption gives . Theorem 2 and the assumption of finiteness of can then be used for Bayesian sampling more generally as follows: Sample from a law proportional to the measure and return . The latter proof does not rely on the existence of a density for with respect to Lebesgue measure. This argument is used in Section 4.1 to provide a concrete example where a traditional Bayesian sampling recipe fails, but the fiducial algorithm from Theorem 2 can be used.
Consider finally frequentist inference. If where has a unique maximum at , then is the maximum likelihood estimator. Assume that the expected value exists. It follows then that the expected value of the fiducial distribution is the shift equivariant estimator with smallest mean square error (Taraldsen and Lindqvist, 2013). It is in particular better than the maximum likelihood estimator when both exist, it is unbiased, and the standard error is given by the standard deviation of .
The fiducial distribution is also a confidence distribution since is a pivotal. Consequently an expanded uncertainty can be found corresponding to confidence intervals. Symmetric, shortest, or uniformly most powerful limits can be calculated. The most powerful limits follow with reference to the likelihood ratio test as exemplified for the exponential by Taraldsen (2011). This reference also gives the route for the inclusion of the effect of finite resolution into the analysis.
The previous analysis with the assumption can be generalized verbatim to the case where
is a finite dimensional real or complex vector space. The property
holds for the finite dimensional Lebesgue distribution for . The further generalization to the case where is an infinite dimensional Hilbert space gives an example where the Bayesian algorithm fails to produce optimal frequentist inference. The fiducial argument given above holds also for the infinite dimensional case, and gives optimal inference as stated above (Taraldsen and Lindqvist, 2013).The analysis can be generalized further to the case . This includes in particular the case of a random sample of size from the original model given in equation (4), but the independence assumption is not required in the following argument. Equation (4) must be replaced by the equation corresponding to the first component of the random element in . The law must be replaced by the conditional law . Except for the practical difficulties related to this conditional law, the analysis proceeds as before. Optimal frequentist inference procedures including confidence distributions follow from the resulting fiducial also in this case (Taraldsen and Lindqvist, 2013).
4 Location examples
4.1 A singular example
The purpose of this example is to demonstrate that Theorem 2 can be used to calculate the Bayesian posterior in certain cases where the traditional Bayesian recipe fails.
Let the Monte Carlo law give probability to the value for . The model given by with gives a law which is concentrated on . The traditional Bayesian posterior would usually be calculated by , but this fails here since the density fails to exist for the case considered.
Consider next the algorithm given by Theorem 2. The relation gives the following recipe: Sample from a law that gives relative probability to the values and . The resulting is a sample from the Bayesian posterior . The conclusion is that the posterior gives probability to the two values for .
The uniform prior case gives that the posterior equals the fiducial which gives probability to .
4.2 Normal distribution
Assume that where the Monte Carlo law of corresponds to a random sample of size
from the standard normal distribution. The
is then a fiducial model for a random sample of size from awhere the variance
is assumed known.This gives with and has a standard normal distribution. The is then a fiducial model for the sufficient statistic which is a statistical model.
The fiducial based on the sufficient statistic is the law of which is . This gives the optimal equivariant estimator , the standard error , and the expanded error where the coverage factor gives the level .
The Bayesian conclusion with the uniform law as prior is given by the same numbers since the fiducial coincides with the posterior in this case.
4.3 Gamma distribution I
The example here is a generalization of the case given by a random sample from the exponential distribution
(Taraldsen and Lindqvist, 2013). Let , whereis the inverse CDF of the gamma distribution with scale
. If independent, then the inversion method gives that is a fiducial model for a random sample from the gamma density:(5) 
It follows from this density that is sufficient. A fiducial model from the above fiducial model is then , where the Monte Carlo variable has a distribution.
The fiducial equation gives the fiducial with an distribution. This is a confidence distribution for , and also the Bayesian posterior corresponding to a uniform prior for . The mean
(6) 
is the best Bayesian estimator for the quadratic loss. It can be seen as a sample size adjustment of the likelihood estimate .
The scale model transforms to the location model . The best equivariant estimator for is then , and this integral equals where is the digamma function.
The best equivariant estimate for is
(7) 
This is best with respect to the squared distance from the Fisher metric as explained in more detail by Taraldsen and Lindqvist (2013).
The reason for the choice of the above formulation of equation (7) is that is the uniformly minimum variance estimator of , and the term can be seen as a correction of this. The following asymptotic and divergent series for shows in particular consistency of the estimator in equation (7) with the more common estimator in the limit of infinite sample size .
The main reason for the inclusion of this example is not the possibly novel result given by equation (7), but rather demonstration purposes. We consider the arguments as given above as a competitive alternative to the arguments given by a Bayesian calculation. The Bayesian calculation is of course possible in this case, but it seems more cumbersome to us. The claims on optimality can indeed also be proved directly without any mention of a Bayesian prior (Taraldsen and Lindqvist, 2013).
5 Onedimensional fiducial inference
Fiducial inference was first considered in the onedimensional case. This is discussed here, and the connection between the original definition and the more general Definition 3 is in particular explained.
5.1 Lindley’s result
Lindley (1958) considered the onedimensional case. His sufficient and necessary condition for a fiducial posterior is equivalent with the conditions given in Theorem 1
. The result is only valid by consideration of a more restrictive definition of the fiducial distribution defined directly and uniquely by an absolutely continuous cumulative distribution function. This is explained next.
The monotonicity of a simple fiducial model has as a consequence monotonicity in of the cumulative distribution function of . In the following it is furthermore assumed that is absolutely continuous and onto . The relation can be inverted to give and . The well known inversion method gives that with the uniform law on is a fiducial model for . This is the Fisher fiducial model, and it is a simple conventional fiducial model. It can be shown that the corresponding Fisher fiducial distribution coincides with the fiducial distribution of the original fiducial model (Dawid and Stone, 1982). Fiducial inference is hence unique in this case. If the cumulative distribution is decreasing in , then is the cumulative fiducial distribution.
The result of Lindley is that a fiducial posterior is obtained if and only if the fiducial model is a transformation of a fiducial model where . The prior for is Lebesgue measure on and the resulting fiducial model is the location model. The transformation assumption is that with and .
The if part of the claim is a special case of the results discussed in Section 3 since both Bayesian and fiducial inference behave consistently under transformations. The if part does not require existence of densities, and this result here is then an extension of the results of Lindley.
An example which is more general than the Lindley case is obtained by choosing a which is strictly increasing, but nowhere differentiable. The result is then a singular continuous fiducial posterior, and this is not covered by the proof of Lindley. Another class of examples is given by choosing an arbitrary probability distribution which does not need to be absolutely continuous. A third class of examples not covered by Lindley is given by countable equipped with a possibly noncommutative group or loop operation.
It remains to prove the only if part of the fiducial posterior claim given the above restrictions on the cumulative distribution. The necessary parts of the argument of Lindley is reproduced next.
Assume that the fiducial model has a fiducial posterior in the sense that the fiducial density equals the posterior density . This gives the following generalization of the oneway wave equation
(8) 
A general solution is given by , where and . Consequently, the family of conditional distributions for is a location family with location parameter . The oneone correspondence proves that is given by a transformation of the location group model.
A particularly nice aspect of the above proof is that it gives explicitly the required transformation to a standard location model. The function is the cumulative distribution of the (marginal) law of , and a fiducial posterior is obtained if and only if the variable corresponds to a standard location model.
It is also the explicit transformation that ensures that the law of is the uniform law on the real line:
(9) 
A particular consequence is that the prior law of is always improper when the posterior coincides with the fiducial.
5.2 The correlation coefficient
Let be the cumulative distribution function of the empirical correlation coefficient of a random sample from the bivariate normal distribution. The parameter is the correlation coefficient. A fiducial model is given by a uniform law and the fiducial relation . It is possible to sample from the fiducial based on , but a much simpler algorithm is described in section 6.2 below.
In this case it is known that there exists no prior on that gives the fiducial as a posterior (Berger and Sun, 2008, p.966). The proof is not trivial. The fiducial for the correlation coefficient gives the very first example (Fisher, 1930) of a derivation of a fiducial distribution (Fisher, 1973, p.176). The fiducial for the correlation coefficient is, however, a Bayesian posterior from the multivariate normal model considered in section 6.2.
5.3 Gamma distribution II
Consider a random sample from the gamma density
(10) 
The case with a general scale can be reduced to the case of a scale by consideration of . It will hence initially be assumed that .
The form of the density shows that is sufficient. A fiducial model is given by where is the inverse CDF of the gamma distribution with scale and independent, Each is increasing in , since is increasing.
An alternative fiducial model is given by where is the CDF of and . An explicit expression for can be given in terms of the Meijer Gfunction using results by Nadarajah (2011). Both models are simple, and give the same fiducial distribution. We conjecture that this fiducial is not obtainable as a Bayesian posterior, but do not attempt a proof.
The fiducial is a confidence distribution for the shape , and both of the previous fiducial models give sampling algorithm. Reasonable estimators for are given by and corresponding to a squared distance loss on the direct and logarithmic scale respectively. Alternatives are given by the Fisher information metric or an entropy distance. Natural competitors are the maximumlikelihood and the Jeffreys prior Bayesian versions of the previous fiducial estimators. A detailed discussion of this will not be give here.
6 Group and loop models
It will next be explained, as promised in the abstract, that Fraser’s result on fiducial posteriors follows as a special case of Theorem 1.
6.1 A generalized locationscale model
Let be defined by
(11) 
where are columns of length and is a lower triangular matrix with positive diagonal. The case gives the standard locationscale model with , and equation (11) can be seen as a natural generalization.
The generalized locationscale group with elements is discussed in more detail by Fraser (1979, p.175) in the context of structural inference. Multiplication is defined by , the inverse is , and the identity is . The group may also be identified with the group of lower triangular matrices on the block form
(12) 
which gives the previous calculation rules from matrix multiplication directly.
A Monte Carlo law gives that is a fiducial model for the conditional law of . It will be assumed that the are independent and corresponds to a random sample of size from a known probability distribution on . The columns corresponds then also to a random sample of size from a distribution on with and , where it is assumed that and .
The result so far is a fiducial model where the model parameter space corresponds to the mean and covariance of some multivariate law on . It should be observed that the Cholesky decomposition determines uniquely, and it is hence a matter of choice if or is considered as a model parameter.
The model given by equation (11) is not a simple fiducial model, but it can be reduced to a simple fiducial model by conditioning similarly to how the location model was treated. The general recipe for this is explained by Taraldsen and Lindqvist (2013), but the details of this will not be give here.
6.2 The multivariate normal
The possibly most important group model is given from the previous discussion and assuming that is the law of a matrix of independent standard normal variables. The result is then a fiducial model corresponding to a random sample of size from the multivariate normal . This is not the only possible fiducial model for this case, but other possibilities will not be discussed.
A simple fiducial model is then obtained from the sufficient statistic where is the Cholesky decomposition of the empirical covariance matrix of . The fiducial model from equation (11) gives then a fiducial model for where
(13) 
and . This model is simple, and the fiducial as given by Definition 3 is the law of .
Let be the right Haar prior on . The explicit form for this is not needed in the following argument. The right invariance gives , and Theorem 1 gives that the Bayesian posterior coincides with the fiducial.
Sampling from the posterior can be done by alternative methods, but it seems that the algorithm that follows from the fiducial argument is the simplest possible that generate independent samples. It involves only standard matrix calculations including solving lower triangular linear systems, and calculation of Cholesky decompositions. This gives in particular a simple sampling algorithm for the fiducial distribution of the correlation coefficient considered in section 5.2.
6.3 General group case
Assume that is given by group multiplication. Let be a right Haar measure on the group . It follows from the right invariance that for all . Theorem 1 can now be applied, and it follows that the distribution of conditional on equals both the fiducial and the posterior. This case can be referred to as the fiducial group model case . The fiducial model for is pivotal, and the right Haar distribution is a matching prior: The posterior is fiducial and also a confidence distribution since gives a pivotal quantity. This result is the result obtained by Fraser (1961b), but he obtained it by a different argument.
The previous group case is important since it gives a multitude of nontrivial examples where the assumptions in Theorem 1 are fulfilled. It is in particular noteworthy that the required finiteness of and follows as consequences in the fiducial group model.
Existence of a finite random quantity such that the distribution does not depend on is a nontrivial problem in general. It is a generalization of the existence and uniqueness problem for Haar measure on a group. This is ensured in the fiducial group case if it is assumed that is a locally compact group (Halmos, 1950).
A more general family of examples can be constructed as follows. Let , , , and where the product is the group multiplication in a group equipped with a right Haar measure . Assume that is such that . It follows then that . If and are invertible, then the fiducial model is simple and the fiducial posterior is distributed like conditional on . It can be observed that the functions and can be used to identify and respectively both as sets and as groups. The previous model is hence essentially reducible to the group case by a change of variables.
The reduction can alternatively be formulated as follows. Define a new parameter , a new variable , and let . It follows that is a fiducial group model for , and inference can be based on this. The result is the same as in the previous paragraph. It can not be concluded that the original fiducial model is a group model, but the model is transformed into a fiducial group model .
Let be a group with an invariant measure, and let with oneone on . This is a special case of the case considered in the previous two paragraphs. This defines a binary operation on which need not be a group since the associative law may fail. An example is given by with addition on the real line . It is however a quasigroup (Smith, 2006), but in the context here it is essentially reduced to the group case by relabeling as explained in the previous paragraphs.
6.4 Loop models
A quasigroup is a set equipped with a binary operation such that for each , there exist unique elements such that and . A loop is a quasigroup with an identity element, that is, an element such that for all . A group is a loop so that the associative law holds for all . The concept of a loop within abstract algebra as just defined is probably less familiar to most readers than the concept of a group. On an intuitive level it can be considered to be an object similar to a group, but without the associative law. It will next be explained that loops occur naturally in the context of fiducial theory.
Consider the case where is a bijection separately in both arguments. The fiducial model is then said to be a pivotal and simple model. The bijections defined by can be used to define a change of variables so that it may be assumed that . The result is a set equipped with a binary operation with inverse and inverse . is then a quasigroup. The notation , , and with right and left division is standard. The change of variables can also be chosen so that there is an identity element such that for all . is then a loop. The conclusion is that a change of variables reduces a pivotal simple model to a loop model.
Two examples of loops which do not seem to be essentially reduced to the group case are given next. One example is given by where and are positive definite matrices. This gives an example of a Bruck loop. Another example is where the multiplication is the multiplication of the invertible octonions, which is a Moufang loop. It is not known to the authors if there exist invariant loop measures for certain classes of loops, or for these concrete examples. This would provide examples beyond the group case.
The finite and countable loop cases are trivial in that counting measure is the unique invariant measure, but they can otherwise be quite exotic objects. They do, however, provide examples where the fiducial equals the Bayesian posterior, and this does not follow from the results of Lindley and Fraser.
The onedimensional case considered in Section 5.2 provides in particular an example of a loop on the real line that does not possess an invariant measure. This could be of independent interest, and is hence stated separately here as a Theorem.
Theorem 3.
There exist a smooth loop where an invariant right measure does not exist.
Proof.
The result of Lindley can be reformulated to give an alternative characterization of loops with an invariant measure: A smooth loop on the real line has an invariant measure if and only if it can be reduced to a group by a change of variables. It is unclear if this can be generalized to more general loops. The term smooth is here interpreted to mean infinitely differentiable with continuous derivatives.
7 A fundamental lemma
Lemma 1.
Assume that is finite with and for fixed , where and are finite measures and is jointly measurable. Let . It follows then that
(14) 
Proof.
Change variables from to
∎
The key ingredients in the above proof are the Fubini theorem together with the general changeofvariables theorem when . This theorem is usually proved in the context of probability spaces (Halmos, 1950, p.163, Theorem C), but the proof is also valid for the more general case where is assumed to be finite.
The main point of Lemma 1 is that it provides an explicit expression for the density of , and it follows in particular that this density exists. It follows from the proof that is finite, and that is finite if and only if for a.e. . This condition can be checked in applications.
Proof.
Theorem 1 The assumption gives for a finite measure . The finiteness follows since is a bijection. Lemma 1 gives that is the joint density of , and then also that is finite. A sample from can generally be obtained by sampling from followed by sampling from . The result is identical with the result from the fiducial as defined in Definition 3 since is the density of . ∎
Proof.
Theorem 2 The proof is as the previous, but the density of is now given by a density proportional to . The required normalization is possible since it is assumed that is finite. ∎
8 Closing remarks
As explained in the introduction it is important to establish cases where the fiducial equals a Bayesian posterior, and also the cases where the fiducial is not a Bayesian posterior. In general and special cases this is a difficult task. Theorem 1 shows that existence of a law such that has a law that does not depend on implies that the fiducial equals the resulting Bayesian posterior. The proof of existence of an invariant law is difficult, but in the case of groups the theory is well established. The results of Fraser on fiducial posteriors follow then as corollaries of Theorem 1 as explained in section 6.3.
The correlation coefficient case gives an example where the fiducial is not equal to a Bayesian posterior from the statistical model for the empirical correlation coefficient. We believe that the gamma with known scale, and the gamma where both scale and shape are unknown give two more examples where the fiducial is not a Bayesian posterior, but we do not have a proof of this.
The use of sufficient statistics for the gamma model gives examples of respectively a one and a twodimensional loop model. The question of existence of invariant measures for quasigroups or loops has here been shown to be related to the question of fiducial posteriors. Unfortunately, it seems that the question of existence of invariant measures for quasigroups is an open and difficult question. A byproduct of the discussion given here is Theorem 3 that shows existence of a smooth loop where an invariant right measure does not exist.
Theorem 2 has a more direct application. It gives an alternative algorithm for Bayesian posterior sampling based on a fiducial model.
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