Strong sufficiency

The strong sufficiency is in mathematical statistics a modification of the sufficiency and thus important to answer the question of whether information can be compressed without loss. As in the case of (ordinary) sufficiency, one first defines the strongly sufficient σ-algebra in order to then define the strongly sufficient statistics . Severe sufficiency and sufficiency are related, but are generally not identical. The strong sufficiency goes back to a work by David Blackwell from 1951, see section literature.

definition

Given a statistical model and a σ-algebra with . Then a strongly sufficient σ-algebra for , if a Markov kernel exists from to , is called such that ${\displaystyle (\Omega ,{\mathcal {A}},{\mathcal {P}})}$ ${\displaystyle {\mathcal {S}}}$ ${\displaystyle {\mathcal {S}}\subset {\mathcal {A}}}$ ${\displaystyle {\mathcal {S}}}$ ${\displaystyle {\mathcal {P}}}$ ${\displaystyle \kappa }$ ${\displaystyle (\Omega ,{\mathcal {S}})}$ ${\displaystyle (\Omega ,{\mathcal {A}})}$

${\displaystyle \kappa (\omega ,A)=P(A|{\mathcal {S}})(\omega )=\operatorname {E} _{P}(\mathbf {1} _{A}|{\mathcal {S}})(\omega )}$

for everyone and everyone . ${\displaystyle A\in {\mathcal {A}}}$ ${\displaystyle P\in {\mathcal {P}}}$

A statistic is called a strongly sufficient statistic if the σ-algebra generated by the statistic is a strongly sufficient σ-algebra. ${\displaystyle T}$ ${\displaystyle \sigma (T)}$

Relationship to sufficiency

On borel rule areas strong sufficiency and sufficiency coincide. Because if sufficiency is strong, it is precisely that of the choice of independent -measurable function that is required in the definition of sufficiency. Conversely, if a Borel space is sufficient, the Markov kernel required in the definition of severe sufficiency always exists; it is precisely the kernel that defines the regular conditional distribution that always exists on Borel spaces. ${\displaystyle {\mathcal {S}}}$ ${\displaystyle \kappa (\cdot ,A)=f_{A}}$ ${\displaystyle P\in {\mathcal {P}}}$ ${\displaystyle {\mathcal {S}}}$ ${\displaystyle {\mathcal {S}}}$

use

Strong sufficiency is used, for example, in decision theory. Here decisions are modeled using Markov kernels, the so-called randomized decision functions . In order to measure the damage to a decision function , a risk function is defined, which measures the risk for a decision given a given but unknown parameter and a given decision function. If there is now a strongly sufficient σ-algebra, the decision function can be defined on instead of the large σ-algebra without the risk function changing. The highly sufficient σ-algebra thus already contains all information necessary for the risk function. ${\displaystyle {\mathcal {S}}}$ ${\displaystyle {\mathcal {A}}}$ ${\displaystyle {\mathcal {S}}}$

literature

• Ludger Rüschendorf: Mathematical Statistics . Springer Verlag, Berlin Heidelberg 2014, ISBN 978-3-642-41996-6 , doi : 10.1007 / 978-3-642-41997-3 .
• David Blackwell: Comparison of Experiments , Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, University of California Press, Berkeley 1951, 93-102.