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 .
Distribution-free σ-algebra
If a σ-algebra, then a distribution-free σ-algebra is called with respect to , if
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
- .
The probability measures cannot therefore be differentiated on the basis of their values .
Distribution-free statistics
A statistic
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.
Important statements
Basu's three sentences establish a connection between the concepts of freedom of distribution, sufficiency and completeness . In short they read:
- A sufficient limited complete statistic and a non-distribution statistic are stochastically independent for all .
- If σ-algebras are independent of each other and are sufficient for all , then (under certain additional assumptions) distribution-free.
- Let the σ-algebras be stochastically independent for all and be free of distribution. If then , then is sufficient.
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 .