View source for Fiducial distribution

<!--
f0400801.png
$#A+1 = 63 n = 0
$#C+1 = 63 : ~/encyclopedia/old_files/data/F040/F.0400080 Fiducial distribution
Automatically converted into TeX, above some diagnostics.
Please remove this comment and the {{TEX|auto}} line below,
if TeX found to be correct.
-->

{{TEX|auto}}
{{TEX|done}}

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 [[#References|[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. [[#References|[2]]]–[[#References|[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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  R.A. Fisher,   "Inverse probability"  ''Proc. Cambridge Philos. Soc.'' , '''26'''  (1930)  pp. 528–535</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  D.A.S. Fraser,   "The fiducial method and invariance"  ''Biometrika'' , '''48'''  (1961)  pp. 261–280</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  G.P. Klimov,   "Invariant inferences in statistics" , Moscow  (1973)  (In Russian)</TD></TR></table>

====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., [[#References|[a1]]] for a recent survey.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.G. Pedersen,   "Fiducial inference"  ''Internat. Stat. Rev.'' , '''46'''  (1978)  pp. 147–170</TD></TR></table>