# 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)