Fiducial distribution
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) |
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 |
Fiducial distribution. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fiducial_distribution&oldid=46918