Maximum invariant statistics

A maximally invariant statistic is a special mapping in mathematical statistics . Maximum invariant statistics play an important role in the reduction through invariance and the construction of optimal invariant estimators.

definition

Given is a (multiplicative) group and a set ${\displaystyle ({\mathcal {G}},\cdot )}$

${\displaystyle {\mathcal {U}}=\{u_{\gamma }\mid \gamma \in {\mathcal {G}}\}}$

of measurable transformations on the measuring room . This means that ${\displaystyle (X,{\mathcal {A}}_{X})}$

• Any of the functions

${\displaystyle u_{\gamma }\colon (X,{\mathcal {A}}_{X})\to (X,{\mathcal {A}}_{X})}$

is bijective and can be measured .

• The image

${\displaystyle \gamma \mapsto u_{\gamma }}$

is a group homomorphism from to , provided with the composition of functions . So it applies to everyone and everyone ${\displaystyle ({\mathcal {G}},\cdot )}$ ${\displaystyle {\mathcal {U}}}$ ${\displaystyle \circ }$ ${\displaystyle \gamma _{1},\gamma _{2}\in {\mathcal {G}}}$ ${\displaystyle x\in X}$

${\displaystyle u_{\gamma _{1}}(u_{\gamma _{1}}(x))=u_{\gamma _{1}\cdot \gamma _{2}}(x)}$.

Be another measuring room. Then it is called a measurable function ${\displaystyle (Y,{\mathcal {A}}_{Y})}$

${\displaystyle T\colon (X,{\mathcal {A}}_{X})\to (Y,{\mathcal {A}}_{Y})}$

a maximally invariant statistic if:

• ${\displaystyle T}$is invariant, that is, it applies to all and all (or all ). ${\displaystyle T(u_{\gamma }(x))=T(x)}$ ${\displaystyle x\in X}$ ${\displaystyle \gamma \in {\mathcal {G}}}$ ${\displaystyle u_{\gamma }\in {\mathcal {U}}}$
• Are such that is true, then there exists a (or a ), so that is. ${\displaystyle x_{1},x_{2}\in X}$ ${\displaystyle T(x_{1})=T(x_{2})}$ ${\displaystyle \gamma \in {\mathcal {G}}}$ ${\displaystyle u_{\gamma }\in {\mathcal {U}}}$ ${\displaystyle x_{1}=u_{\gamma }(x_{2})}$

example

As an example, consider and . Let the group be the real numbers , provided with the addition as a link. For define the bijective, measurable mapping ${\displaystyle (X,{\mathcal {A}}_{X})=(\mathbb {R} ^{n},{\mathcal {B}}(\mathbb {R} ^{n}))}$ ${\displaystyle (Y,{\mathcal {A}}_{Y})=(\mathbb {R} ,{\mathcal {B}}(\mathbb {R} ))}$ ${\displaystyle {\mathcal {G}}}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle +}$ ${\displaystyle \gamma \in \mathbb {R} }$

${\displaystyle u_{\gamma }(x)=x+\gamma \mathbf {1} }$.

Here denotes the one vector . The mapping thus shifts each vector by along the diagonal. ${\displaystyle \mathbf {1} }$ ${\displaystyle u_{\gamma }}$ ${\displaystyle x}$ ${\displaystyle \gamma }$

If one denotes the arithmetic mean of the vector , then a maximally invariant statistic is given by ${\displaystyle {\overline {x}}}$ ${\displaystyle x}$

${\displaystyle T(x)=x-{\overline {x}}\cdot \mathbf {1} }$.

Because the arithmetic mean is displacement equivariant, so it is fulfilled

${\displaystyle {\overline {x+\gamma \mathbf {1} }}={\overline {x}}+\gamma }$,

from what

${\displaystyle T(u_{\gamma }(x))=T(x+\gamma \mathbf {1} )=x+\gamma \mathbf {1} -({\overline {x}}+\gamma )\mathbf {1} =x-{\overline {x}}\cdot \mathbf {1} =T(x)}$

results. So is invariant. Well , so is ${\displaystyle T}$ ${\displaystyle T(x_{1})=T(x_{2})}$

${\displaystyle x_{1}-{\overline {x}}_{1}\cdot \mathbf {1} =x_{2}-{\overline {x}}_{2}\cdot \mathbf {1} }$,

from what

${\displaystyle x_{1}=x_{2}+({\overline {x}}_{1}-{\overline {x}}_{2})\mathbf {1} }$

results. This corresponds exactly to a shift of um along the diagonal. Thus applies . So is maximally invariant. ${\displaystyle x_{2}}$ ${\displaystyle {\tilde {\gamma }}=({\overline {x}}_{1}-{\overline {x}}_{2})}$ ${\displaystyle x_{1}=u_{\tilde {\gamma }}(x_{2})}$ ${\displaystyle T}$

properties

Behavior on orbits

Denote

${\displaystyle {\mathcal {O}}_{x^{*}}:=\{x\in X\mid {\text{ es existiert ein }}\gamma \in {\mathcal {G}},{\text{ so dass }}u_{\gamma }(x^{*})=x\}}$

the orbit of , i.e. the set of all elements that results from through group operations. Then the invariance of means that is constant on a given orbit. So are , so is . ${\displaystyle x^{*}}$ ${\displaystyle x^{*}}$ ${\displaystyle T}$ ${\displaystyle T}$ ${\displaystyle x_{1},x_{2}\in {\mathcal {O}}_{x^{*}}}$ ${\displaystyle T(x_{1})=T(x_{2})=T(x^{*})}$

Conversely, the second criterion in the definition means that the orbits can be uniquely identified by the function values ​​of . The level sets of T ${\displaystyle T}$

${\displaystyle {\mathcal {N}}_{y}=\{x\in X\mid T(x)=y\}}$

are clearly defined orbits (or empty).

Generation of invariant statistics

Maximum invariant statistics are maximum in the sense that they generate all further invariant statistics. Specifically, this means that if

${\displaystyle T\colon (X,{\mathcal {A}}_{X})\to (Y,{\mathcal {A}}_{Y})}$

is a maximally invariant statistic and

${\displaystyle S\colon (X,{\mathcal {A}}_{X})\to (Z,{\mathcal {A}}_{Z})}$

an invariant statistic, a function exists

${\displaystyle h\colon Y\to Z}$,

for the

${\displaystyle S=h(T)}$

applies. Every invariant statistic is thus the composition of a maximally invariant statistic and a further function . However, the function is generally not measurable. ${\displaystyle h}$ ${\displaystyle h}$

literature

• Ludger Rüschendorf: Mathematical Statistics . Springer Verlag, Berlin Heidelberg 2014, ISBN 978-3-642-41996-6 , p. 250-254 , doi : 10.1007 / 978-3-642-41997-3 .
• Jürgen Hedderich, Lothar Sachs : Applied statistics . Collection of methods with R. 15th edition. Springer Spectrum, Berlin Heidelberg 2016, ISBN 978-3-662-45690-3 , p. 198-204 , doi : 10.1007 / 978-3-662-45691-0 .