View source for Fiducial distribution

A distribution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f0400801.png" /> of the parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f0400802.png" /> from a family of distributions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f0400803.png" /> of an observation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f0400804.png" />. It was introduced by R.A. Fisher [[#References|[1]]] for numerical <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f0400805.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f0400806.png" /> in the case when the distribution function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f0400807.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f0400808.png" /> decreases as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f0400809.png" /> increases in such a way that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f04008010.png" />, considered as a function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f04008011.png" /> for fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f04008012.png" />, has the properties of a distribution function (in such a situation one often makes use of a sufficient statistic in the role of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f04008013.png" />).

A fiducial distribution is defined for invariant families of distributions (cf. [[#References|[2]]]–[[#References|[4]]]). Namely, suppose that a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f04008014.png" /> of transformations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f04008015.png" /> acts on the sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f04008016.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f04008017.png" />. A family of distributions is called invariant if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f04008018.png" /> has the distribution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f04008019.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f04008020.png" /> has the distribution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f04008021.png" />. In this case one considers equivariant decision rules <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f04008022.png" /> (i.e. such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f04008023.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f04008024.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f04008025.png" />) and invariant loss functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f04008026.png" /> (i.e. such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f04008027.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f04008028.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f04008029.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f04008030.png" />). If the action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f04008031.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f04008032.png" /> is transitive, then the family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f04008033.png" /> has a certain property of homogeneity: For a fixed parameter value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f04008034.png" /> and an observation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f04008035.png" /> with the distribution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f04008036.png" />, the distribution of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f04008037.png" /> runs through the whole family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f04008038.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f04008039.png" /> runs through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f04008040.png" />. Suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f04008041.png" /> is a set of probability measures on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f04008042.png" /> (it is assumed that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f04008043.png" />-algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f04008044.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f04008045.png" /> are given such that the transformations in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f04008046.png" /> are measurable). Let the action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f04008047.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f04008048.png" /> be given by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f04008049.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f04008050.png" />. The fiducial distribution is described by the family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f04008051.png" /> of probability measures on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f04008052.png" /> that minimize the risk <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f04008053.png" /> in the class of equivariant decision rules for every invariant loss function satisfying the following condition of unbiasedness type

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f04008054.png" /></td> </tr></table>

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f04008055.png" /> acts transitively on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f04008056.png" />, then the family of the fiducial distributions is uniquely distinguished by the requirements that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f04008057.png" /> is invariant and that the probable and fiducial probabilities are equal, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f04008058.png" />, for invariant families <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f04008059.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f04008060.png" /> is called invariant if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f04008061.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f04008062.png" /> imply that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040080/f04008063.png" />).

====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>