Equivariant estimator

An equivariant estimator is a special point estimator in estimation theory , a branch of mathematical statistics . In the simplest case, equivariant estimators are characterized by the fact that a transformation of the data leads to an identical transformation of the estimated value. If you shift the data by a certain amount, the estimated value is also shifted by this value.

For equivariate estimators, some optimality conditions can be shown more easily. For example, under certain additional assumptions, locally minimal equivariant estimators are always consistently best unbiased estimators . Important equivariant estimators are the Pitman estimators .

definition

A statistical model is given . Be a group of bijective, measurable transformations from to and let it apply to everyone ${\displaystyle (X,{\mathcal {A}},(P_{\vartheta })_{\vartheta \in \Theta })}$ ${\displaystyle \Theta \subset \mathbb {R} ^{n}}$ ${\displaystyle {\mathcal {Q}}}$ ${\displaystyle (X,{\mathcal {A}})}$ ${\displaystyle (X,{\mathcal {A}})}$ ${\displaystyle q\in {\mathcal {Q}}}$

${\displaystyle q(X)=X{\text{ sowie }}q({\mathcal {A}})={\mathcal {A}}}$.

Then induced over the context ${\displaystyle {\mathcal {Q}}}$

${\displaystyle P_{\vartheta }(A)=P_{{\overline {q}}(\vartheta )}(q(A))}$

a group on . ${\displaystyle {\overline {\mathcal {Q}}}=\{{\overline {q}}\}}$ ${\displaystyle \Theta }$

Furthermore, be

${\displaystyle {\overline {q}}(\Theta )=\Theta {\text{ für alle }}{\overline {q}}\in {\overline {\mathcal {Q}}}.}$

Then is called a point estimator

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

an equivariant estimator if

${\displaystyle d(q(x))={\overline {q}}(d(x)){\text{ für alle }}q\in {\mathcal {Q}}}$

applies.

Equivariant estimator in the location model

Let be a location model, i.e. a statistical model with a location class that is generated by the probability measure . Be ${\displaystyle (\mathbb {R} ^{n},{\mathcal {B}}(\mathbb {R} ^{n}),(P_{\vartheta })_{\vartheta \in \Theta })}$ ${\displaystyle (P_{\vartheta })_{\vartheta \in \Theta }}$ ${\displaystyle P}$

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

the group of translations in along the one vector around . ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle \mathbf {1} }$ ${\displaystyle \vartheta }$

Then applies as required above

${\displaystyle T_{\vartheta }(\mathbb {R} ^{n})=\mathbb {R} ^{n}{\text{ und }}T_{\vartheta }({\mathcal {B}}(\mathbb {R} ^{n}))={\mathcal {B}}(\mathbb {R} ^{n})}$

for everyone . For true even ${\displaystyle \vartheta \in \mathbb {R} }$ ${\displaystyle P_{0}\in (P_{\vartheta })_{\vartheta \in \Theta }}$

${\displaystyle P_{\vartheta }(A)=P_{0}(A-\vartheta \cdot \mathbf {1} )}$,

because lies in the location class. Thus the induced group is given by the translations um . ${\displaystyle P_{0}}$ ${\displaystyle {\overline {\mathcal {Q}}}}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \vartheta }$

Accordingly, a point estimator in this model is an equivariant estimator if and only if ${\displaystyle d}$

${\displaystyle d(x+\mathbf {1} \cdot \vartheta )=d(x)+\vartheta }$

applies.

Equivariant estimator in the scale model

Is a scale model , i.e. a statistical model with a scale family and is ${\displaystyle (\mathbb {R} _{+}^{n},{\mathcal {B}}(\mathbb {R} _{+}^{n}),(P_{\vartheta })_{\vartheta \in (0,\infty )})}$

${\displaystyle {\mathcal {Q}}:=\{S_{\vartheta }\,|\,\vartheta \in (0,\infty )\}{\text{ mit }}S_{\vartheta }(x):=\vartheta \cdot x}$

the group (up ) of the multiplication with a positive real number, then the group (up ) is also the multiplication with a positive real number. Analogous to the above case, this follows via the defining properties of the scale family. Thus, in the scale model, a point estimator is an equivariant estimator if and only if ${\displaystyle \mathbb {R} _{+}^{n}}$ ${\displaystyle {\overline {\mathcal {Q}}}}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle d}$

${\displaystyle d(\vartheta \cdot x)=\vartheta \cdot d(x)}$

is.

Web links

• MS Nikulin: Equivariant estimator . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).

literature

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