Theorems of Basu

The Basu's theorem are three statements of mathematical statistics that a connection between the sufficiency , the completeness and the distribution of free produce.

They were established and proven in 1955 by Debabrata Basu .

sentences

For all sentences there is always a statistical model with a basic set , σ-algebra and distribution class . In addition, let sub- σ-algebras of . ${\displaystyle (\Omega ,{\mathcal {A}},{\mathcal {P}})}$ ${\displaystyle \Omega }$ ${\displaystyle {\mathcal {A}}}$ ${\displaystyle {\mathcal {P}}}$ ${\displaystyle {\mathcal {C}},{\mathcal {D}}}$ ${\displaystyle {\mathcal {A}}}$

Relationship sufficiency, completeness and freedom of distribution

Is

${\displaystyle f:(\Omega ,{\mathcal {A}})\to (X,{\mathcal {X}})}$

a non-distribution statistic and is

${\displaystyle g:(\Omega ,{\mathcal {A}})\to (Y,{\mathcal {Y}})}$

a sufficient and limited complete statistic , then and for all are stochastically independent random variables . ${\displaystyle f}$ ${\displaystyle g}$ ${\displaystyle P\in {\mathcal {P}}}$

Freedom of distribution and independent sufficient σ-algebras

A lot of exist for everyone , so that ${\displaystyle {\underline {P}},{\overline {P}}\in {\mathcal {P}}}$ ${\displaystyle (P_{i})_{i=1,\dots ,n}\in {\mathcal {P}}}$ ${\displaystyle {\underline {P}}=P_{1},P_{2},\dots ,P_{n}={\overline {P}}}$

and and are not singular to each other. If and are stochastically independent σ-algebras for all and is a sufficient σ-algebra , then is a distribution-free σ-algebra . ${\displaystyle P_{i}}$ ${\displaystyle P_{i+1}}$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle {\mathcal {D}}}$ ${\displaystyle P\in {\mathcal {P}}}$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle {\mathcal {D}}}$

Sufficiency of maximum supplements

Be stochastically independent for everyone and be free of distribution. If then , then is sufficient. ${\displaystyle {\mathcal {C}},{\mathcal {D}}}$ ${\displaystyle P\in {\mathcal {P}}}$ ${\displaystyle {\mathcal {D}}}$ ${\displaystyle \sigma ({\mathcal {C}},{\mathcal {D}})={\mathcal {A}}}$ ${\displaystyle {\mathcal {D}}}$

literature

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