Fiducial distribution

A distribution $ P _ {x} ^ {*} $ of the parameter $ \theta $ from a family of distributions $ {\mathcal P} = \{ {P _ \theta } : {\theta \in \Theta } \} $ of an observation $ x $. It was introduced by R.A. Fisher [1] for numerical $ \theta $ and $ x $ in the case when the distribution function $ F ( x \mid \theta ) $ of $ x $ decreases as $ \theta $ increases in such a way that $ F ^ { * } ( \theta | x) = 1 - F ( x | \theta ) $, considered as a function of $ \theta $ for fixed $ x $, has the properties of a distribution function (in such a situation one often makes use of a sufficient statistic in the role of $ x $).

A fiducial distribution is defined for invariant families of distributions (cf. [2]–[4]). Namely, suppose that a group $ G $ of transformations $ g $ acts on the sets $ X $ and $ \Theta $. A family of distributions is called invariant if $ gx $ has the distribution $ P _ {g \theta } $ when $ x $ has the distribution $ P _ \theta $. In this case one considers equivariant decision rules $ \delta : X \rightarrow D $( i.e. such that $ \delta ( gx) = g \delta ( x) $ for all $ x $ and $ g $) and invariant loss functions $ L _ \theta ( d) $( i.e. such that $ L _ {g \theta } ( gd) = L _ \theta ( d) $ for all $ \theta $, $ d $ and $ g $). If the action of $ G $ on $ \Theta $ is transitive, then the family $ {\mathcal P} $ has a certain property of homogeneity: For a fixed parameter value $ \theta _ {0} $ and an observation $ x $ with the distribution $ P _ {\theta _ {0} } $, the distribution of $ gx $ runs through the whole family $ {\mathcal P} $ as $ g $ runs through $ G $. Suppose that $ D $ is a set of probability measures on $ \Theta $( it is assumed that $ \sigma $- algebras $ {\mathcal B} ( \Theta ) $ and $ {\mathcal B} ( X) $ are given such that the transformations in $ G $ are measurable). Let the action of $ G $ on $ D $ be given by $ ( g \alpha ) ( B) = \alpha ( g ^ {-} 1 ( B)) $, $ G \in {\mathcal B} ( \Theta ) $. The fiducial distribution is described by the family $ {\mathcal P} ^ {*} = \{ {P _ {x} ^ {*} } : {x \in X } \} $ of probability measures on $ \Theta $ that minimize the risk $ \int L _ \theta ( \delta ( x)) dP _ \theta ( x) $ in the class of equivariant decision rules for every invariant loss function satisfying the following condition of unbiasedness type

$$ \int\limits L _ \theta ( \alpha ) \ d \beta ( \theta ) \geq \ \int\limits L _ \theta ( \beta ) \ d \beta ( \theta ). $$

If $ G $ acts transitively on $ X $, then the family of the fiducial distributions is uniquely distinguished by the requirements that $ {\mathcal P} ^ {*} = \{ {P _ {x} ^ {*} } : {x \in X } \} $ is invariant and that the probable and fiducial probabilities are equal, $ P _ \theta \{ \theta \in S ( x) \} = P _ {x} ^ {*} \{ \theta \in S ( x) \} $, for invariant families $ S ( x) $( $ S ( x) $ is called invariant if $ \theta \in S ( x) $, $ g \in G $ imply that $ g \theta \in S ( gx) $).

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)

Comments

There has been a continued controversy as to what Fisher really meant by fiducial distributions, many authors finding the idea quite meaningless. See, e.g., [a1] for a recent survey.

References

[a1]	J.G. Pedersen, "Fiducial inference" Internat. Stat. Rev. , 46 (1978) pp. 147–170