# Fiducial distribution

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A distribution of the parameter from a family of distributions of an observation . It was introduced by R.A. Fisher [1] for numerical and in the case when the distribution function of decreases as increases in such a way that , considered as a function of for fixed , has the properties of a distribution function (in such a situation one often makes use of a sufficient statistic in the role of ).

A fiducial distribution is defined for invariant families of distributions (cf. [2][4]). Namely, suppose that a group of transformations acts on the sets and . A family of distributions is called invariant if has the distribution when has the distribution . In this case one considers equivariant decision rules (i.e. such that for all and ) and invariant loss functions (i.e. such that for all , and ). If the action of on is transitive, then the family has a certain property of homogeneity: For a fixed parameter value and an observation with the distribution , the distribution of runs through the whole family as runs through . Suppose that is a set of probability measures on (it is assumed that -algebras and are given such that the transformations in are measurable). Let the action of on be given by , . The fiducial distribution is described by the family of probability measures on that minimize the risk in the class of equivariant decision rules for every invariant loss function satisfying the following condition of unbiasedness type

If acts transitively on , then the family of the fiducial distributions is uniquely distinguished by the requirements that is invariant and that the probable and fiducial probabilities are equal, , for invariant families ( is called invariant if , imply that ).

#### References

 [1] R.A. Fisher, "Inverse probability" Proc. Cambridge Philos. Soc. , 26 (1930) pp. 528–535 [2] D.A.S. Fraser, "The fiducial method and invariance" Biometrika , 48 (1961) pp. 261–280 [3] G.P. Klimov, "On the fiducial approach in statistics" Soviet Math. Dokl. , 11 : 2 (1970) pp. 442–444 Dokl. Akad. Nauk SSSR , 191 : 4 (1970) pp. 763–765 [4] G.P. Klimov, "Invariant inferences in statistics" , Moscow (1973) (In Russian)