Halmos-Savage theorem

The set of Halmos-Savage is a tenet of mathematical statistics , which in the presence of a dominant class distribution a necessary and sufficient criterion for the sufficiency of σ-algebras (and thus also of Statistics supplies). The Halmos-Savage theorem is thus an aid to check whether certain functions enable data compression without loss of information. The Neyman criterion for sufficiency , which is easier to handle, can be derived from the Halmos-Savage theorem . Criteria for the existence of minimally sufficient σ-algebras can also be derived from the set of criteria .

The theorem was proven in 1949 by Paul Halmos and Leonard J. Savage .

Framework

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

For any distribution class, let the set of all -null sets. For a dominated distribution class there is always a dominant one , so that and is a countable convex combination with really positive coefficients of elements . So it applies ${\displaystyle {\mathcal {P}}}$ ${\displaystyle {\mathcal {N}}_{\mathcal {P}}}$ ${\displaystyle {\mathcal {P}}}$ ${\displaystyle P^{*}}$ ${\displaystyle {\mathcal {N}}_{\mathcal {P}}={\mathcal {N}}_{P^{*}}}$ ${\displaystyle P^{*}}$ ${\displaystyle {\mathcal {P}}}$

${\displaystyle P^{*}=\sum _{i=1}^{\infty }\alpha _{i}P_{i}{\text{ mit }}\alpha _{i}>0{\text{ und }}\sum _{i=1}^{\infty }\alpha _{i}=1}$.

statement

Let be a dominated distribution class and as given above. Then a sub-σ-algebra of if and suffizient if for all a function exists so that -almost certainly the radon Nikodým derivation of respect is so ${\displaystyle {\mathcal {P}}}$ ${\displaystyle P^{*}}$ ${\displaystyle {\mathcal {S}}}$ ${\displaystyle {\mathcal {A}}}$ ${\displaystyle P\in {\mathcal {P}}}$ ${\displaystyle f_{P}\in {\mathcal {L}}(X,{\mathcal {S}})}$ ${\displaystyle f_{P}}$ ${\displaystyle P^{*}}$ ${\displaystyle P}$ ${\displaystyle P^{*}}$

${\displaystyle f_{P}={\frac {\mathrm {d} P}{\mathrm {d} P^{*}}}\quad P^{*}{\text{-fast sicher}}}$.

example

Let σ-algebras be and be sufficient. In addition, let it be a dominated distribution class. Then, according to Halmos-Savage's theorem, there exists such that and ${\displaystyle {\mathcal {S}}_{1}\subset {\mathcal {S}}_{2}\subset {\mathcal {A}}}$ ${\displaystyle {\mathcal {S}}_{1}}$ ${\displaystyle {\mathcal {P}}}$ ${\displaystyle f_{P}}$ ${\displaystyle f_{P}\in {\mathcal {L}}(X,{\mathcal {S}}_{1})}$

${\displaystyle f_{P}={\frac {\mathrm {d} P}{\mathrm {d} P^{*}}}\quad P^{*}{\text{-fast sicher}}}$.

But there is, applies . Since the density property is still fulfilled, repeated use of the sentence is also sufficient. ${\displaystyle L(X,{\mathcal {S}}_{1})\subset L(X,{\mathcal {S}}_{2})}$ ${\displaystyle f_{P}\in L(X,{\mathcal {S}}_{2})}$ ${\displaystyle f_{P}}$ ${\displaystyle {\mathcal {S}}_{2}}$

Note that this statement does not generally apply and that this is one of the shortcomings of the term sufficiency.

literature

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

Individual evidence

1. ↑ Halmos, Savage: Application of the Radon-Nikodym Theorem to the Theory of Sufficient Statistics, Annals of Mathematical Statistics, Volume 20, 1949, pp. 225-241, Project Euclid