Freedom of distribution

The distribution of freedom is a concept of mathematical statistics , which formalized that for certain amount systems or using certain measurable maps no information can be extracted, so they are uninformative. Freedom of distribution is thus the counterpart to sufficiency , which formalizes that all relevant data can be extracted. As with sufficiency, a distinction is made between distribution-free σ-algebras and distribution-free statistics .

definition

A statistical model with a distribution class is given . ${\displaystyle (\Omega ,{\mathcal {A}},{\mathcal {P}})}$ ${\displaystyle {\mathcal {P}}}$

Distribution-free σ-algebra

If a σ-algebra, then a distribution-free σ-algebra is called with respect to , if ${\displaystyle {\mathcal {V}}\subset {\mathcal {A}}}$ ${\displaystyle {\mathcal {V}}}$ ${\displaystyle {\mathcal {P}}}$

${\displaystyle P_{i}(V)=P_{j}(V){\text{ für alle }}V\in {\mathcal {V}}{\text{ und alle }}P_{i},P_{j}\in {\mathcal {P}}}$

applies.

If one denotes the restriction of the domain of definition of the probability measure to the σ-algebra , then applies to a distribution-free σ-algebra with respect to ${\displaystyle P|_{\mathcal {V}}}$ ${\displaystyle {\mathcal {V}}}$ ${\displaystyle {\mathcal {P}}}$

${\displaystyle P_{i}|_{\mathcal {V}}=P_{j}|_{\mathcal {V}}{\text{ für alle }}P_{i},P_{j}\in {\mathcal {P}}}$.

The probability measures cannot therefore be differentiated on the basis of their values . ${\displaystyle {\mathcal {V}}}$

Distribution-free statistics

A statistic

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

is called a distribution-free statistic if and only if the σ-algebra generated by is a distribution-free σ-algebra with respect to . Equivalent to that is generated from the statistics picture dimensions of identical all. ${\displaystyle T}$ ${\displaystyle \sigma (T)=T^{-1}({\mathcal {A}}^{*})}$ ${\displaystyle {\mathcal {P}}}$ ${\displaystyle {\mathcal {P}}}$

Important statements

Basu's three sentences establish a connection between the concepts of freedom of distribution, sufficiency and completeness . In short they read:

1. A sufficient limited complete statistic and a non-distribution statistic are stochastically independent for all . ${\displaystyle P\in {\mathcal {P}}}$
2. If σ-algebras are independent of each other and are sufficient for all , then (under certain additional assumptions) distribution-free. ${\displaystyle {\mathcal {C}},{\mathcal {D}}}$ ${\displaystyle P\in {\mathcal {P}}}$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle {\mathcal {D}}}$
3. Let the σ-algebras be stochastically independent for all and be free of distribution. If then , then is sufficient. ${\displaystyle {\mathcal {C}},{\mathcal {D}}}$ ${\displaystyle P\in {\mathcal {P}}}$ ${\displaystyle {\mathcal {D}}}$ ${\displaystyle \sigma ({\mathcal {C}},{\mathcal {D}})={\mathcal {A}}}$ ${\displaystyle {\mathcal {C}}}$

Generalizations

A generalization of a non-distribution statistic is a pivot statistic . These are used in the construction of range estimates and thus in the determination of confidence ranges.

literature

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