Invariant estimator

An invariant estimator is a special point estimator in estimation theory , a branch of mathematical statistics . Invariant estimators are characterized by the fact that their estimated value does not change if the data is transformed in a certain way. Frequently used transformations are, for example, scaling or displacements.

Invariant estimators are used, for example, to study the structure of equivariant estimators .

definition

Let us be given a set of measurable functions from to that form a group with regard to the concatenation of functions . ${\displaystyle {\mathcal {Q}}}$ ${\displaystyle (X,{\mathcal {A}})}$ ${\displaystyle (X,{\mathcal {A}})}$ ${\displaystyle \circ }$

Then it is called a measurable function

${\displaystyle d:(X,{\mathcal {A}})\to (\mathbb {R} ,{\mathcal {B}}(\mathbb {R} ))}$

an invariant estimator (with respect to ) if the following applies to all : ${\displaystyle {\mathcal {Q}}}$ ${\displaystyle q\in {\mathcal {Q}}}$

${\displaystyle d(q(x))=d(x)}$

Examples

It is always . ${\displaystyle (X,{\mathcal {A}})=(\mathbb {R} ^{n},{\mathcal {B}}(\mathbb {R} ^{n}))}$

Translation-invariant estimator

If one chooses the translations um along the one vector as a group , that is ${\displaystyle \vartheta }$ ${\displaystyle \mathbf {1} }$

${\displaystyle T_{\vartheta }(x)=x-\vartheta \cdot \mathbf {1} {\text{ und }}{\mathcal {Q}}=\{T_{\vartheta }\,|\,\vartheta \in \mathbb {R} \}}$,

then an estimator is translation invariant if and only if

${\displaystyle d(x-\vartheta \cdot \mathbf {1} )=d(x){\text{ für alle }}\vartheta \in \mathbb {R} }$

is. Translation invariance is, for example, a requirement that variance estimators have to meet, since the variance as a measure of dispersion should not be dependent on displacements.

Scaling invariant estimator

If you choose the scaling as a group, so

${\displaystyle S_{\vartheta }(x)=\vartheta \cdot x{\text{ und }}{\mathcal {Q}}=\{S_{\vartheta }\,|\,\vartheta \in \mathbb {R} \setminus \{0\}\}}$

so an estimator is scale-invariant if

${\displaystyle d(\vartheta \cdot x)=d(x){\text{ für alle }}\vartheta \in \mathbb {R} \setminus \{0\}}$

is.

Additional terms

Maximum invariant estimator

→ Main article : Maximum invariant statistics

A tightening of invariant estimators are maximally invariant estimators.

An estimator is called maximally invariant if it is invariant, and for every two it holds that if and only then ${\displaystyle d}$ ${\displaystyle x,y\in X}$

${\displaystyle d(x)=d(y)}$

applies if there is one such that ${\displaystyle q\in {\mathcal {Q}}}$

${\displaystyle q(x)=y}$

is.

In the case of maximally invariant estimators, all arguments that assume the same function value are on the same path of the group . Maximum invariant estimators can be found, for example, in the definition of Pitman estimators . ${\displaystyle {\mathcal {Q}}}$

P- almost invariant estimator

A set of probability measures is given and the set of all - zero sets is given . It means a -almost invariant estimator if all one does so ${\displaystyle {\mathcal {P}}}$ ${\displaystyle {\mathcal {N}}_{\mathcal {P}}}$ ${\displaystyle {\mathcal {P}}}$ ${\displaystyle d}$ ${\displaystyle {\mathcal {P}}}$ ${\displaystyle q\in {\mathcal {Q}}}$ ${\displaystyle N\in {\mathcal {N}}_{\mathcal {P}}}$

${\displaystyle d(q(x))=d(x){\text{ für alle }}x\in X\setminus N}$.

${\displaystyle {\mathcal {P}}}$-Almost invariant estimators allow a violation of the invariance property on a null set.

literature

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