Minimal sufficiency

The Minimalsuffizienz is in mathematical statistics a tightening of sufficiency . Sufficiency answers the question of whether a system of quantities contains all relevant information or whether an image transmits all relevant information. The minimum sufficiency then asks for the maximum possible compression of the data, for example for the smallest σ-algebra that contains all information of interest. As with sufficiency, minimal sufficiency is defined for σ-algebras and, based on this, for statistics. The closely related minimally sufficient statistic can coincide with this definition, but generally does not.

definition

A statistical model with a distribution class is given . A sufficient σ-algebra is called a minimally sufficient σ-algebra if it is contained in every other sufficient σ-algebra except for -zero sets , i.e. ${\displaystyle (\Omega ,{\mathcal {A}},{\mathcal {P}})}$ ${\displaystyle {\mathcal {P}}}$ ${\displaystyle {\mathcal {M}}}$ ${\displaystyle {\mathcal {P}}}$ ${\displaystyle {\mathcal {S}}}$

${\displaystyle {\mathcal {M}}\subset {\mathcal {S}}\quad {\mathcal {P}}{\text{-fast sicher für alle suffizienten }}{\mathcal {S}}}$.

If one denotes the set of all -zero sets, then this is equivalent to . ${\displaystyle {\mathcal {N}}_{\mathcal {P}}}$ ${\displaystyle {\mathcal {P}}}$ ${\displaystyle {\mathcal {M}}\subset \sigma ({\mathcal {S}},{\mathcal {N}}_{\mathcal {P}})}$

A statistic is called derived

${\displaystyle T:(\Omega ,{\mathcal {A}})\to (\Omega ^{*},{\mathcal {A}}^{*})}$

minimally sufficient if there is a minimally sufficient σ-algebra. ${\displaystyle \sigma (T)}$

Must be distinguished from the minimalsuffiziente statistics: the statistics is a minimalsuffiziente statistics (even minimally sufficient statistic called) if for any sufficient statistic ${\displaystyle T}$

${\displaystyle {\bar {T}}:(\Omega ,{\mathcal {A}})\to ({\bar {\Omega }},{\bar {\mathcal {A}}})}$

a picture in another measuring room ${\displaystyle ({\bar {\Omega }},{\bar {\mathcal {A}}})}$

${\displaystyle \varphi :{\bar {\Omega }}\to \Omega ^{*}}$

exists so that up to -zero sets. ${\displaystyle T=\varphi \circ {\bar {T}}}$ ${\displaystyle {\mathcal {P}}}$

Remarks

• As already noted, the minimal sufficiency of the σ-algebra generated by a statistic and the fact that the statistic is a minimally sufficient statistic do not always coincide. In Borel spaces , however, both terms are identical. In general, however, linguistic precision is required here in order to prevent misunderstandings.
• In general, there is no minimally sufficient σ-algebra and therefore no statistic whose generated σ-algebra is minimally sufficient.

Existence statements

With dominated distribution classes

If there is a dominated distribution class , then there is a minimally sufficient σ-algebra; it is given by ${\displaystyle {\mathcal {P}}}$

${\displaystyle {\mathcal {M}}=\sigma (f_{P}|f_{P}={\tfrac {dP}{dP^{*}}},P\in {\mathcal {P}})}$.

The minimally sufficient σ-algebra is thus generated from the densities of the probability measures with respect to . Here is a dominant measure that can be represented as a countable convex combination of elements of . The proof is given with Halmos-Savage's theorem . ${\displaystyle P^{*}}$ ${\displaystyle P^{*}}$ ${\displaystyle {\mathcal {P}}}$

With separability of the distribution class

If the distribution class is separable with respect to the norm of total variation , then there are minimally sufficient statistics.

Web links

• AS Kholevo: Minimal sufficient statistics . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).

literature

• Ludger Rüschendorf: Mathematical Statistics . Springer Verlag, Berlin Heidelberg 2014, ISBN 978-3-642-41996-6 , doi : 10.1007 / 978-3-642-41997-3 .