Location class

A location class , also known as a location family , translation class or translation family , is a special distribution class in mathematical statistics . Location classes are clearly created by shifting a given probability distribution by a certain value. The set of all these shifted probability distributions then forms the location class. A stochastic model whose distribution class is a location class is called a location model . Location classes are used, for example, in the investigation of equivariant estimators and translation invariant estimators and belong to the Q-invariant distribution classes .

definition

On the real numbers

A probability distribution is given . Define ${\displaystyle P}$ ${\displaystyle (\mathbb {R} ,{\mathcal {B}}(\mathbb {R} ))}$

${\displaystyle P_{\vartheta }(B):=P(B-\vartheta ){\text{ für }}\vartheta \in \mathbb {R} }$

or equivalent to the Dirac distribution in ${\displaystyle \delta _{\vartheta }}$ ${\displaystyle \vartheta }$

${\displaystyle P_{\vartheta }:=(P*\delta _{\vartheta })}$.

Here denotes the convolution of the probability measures. Then is called ${\displaystyle *}$

${\displaystyle {\mathcal {P}}_{L}:=\{P_{\vartheta }\,|\,\vartheta \in \mathbb {R} \}}$

the location class generated by. ${\displaystyle P}$

In higher dimensions

For a probability measure on one defines ${\displaystyle P}$ ${\displaystyle (\mathbb {R} ^{n},{\mathcal {B}}(\mathbb {R} ^{n}))}$

${\displaystyle P_{\vartheta }(B):=P(B-\mathbf {1} \cdot \vartheta )}$,

where the one vector called, so a vector in all ones as entries. Analogous to above then means ${\displaystyle \mathbf {1} }$ ${\displaystyle \mathbb {R} ^{n}}$

${\displaystyle {\mathcal {P}}_{L}:=\{P_{\vartheta }\,|\,\vartheta \in \mathbb {R} \}}$

the location class generated by. ${\displaystyle P}$

example

Let be a standard normal distribution , i.e. in distribution. Then ${\displaystyle P}$ ${\displaystyle P={\mathcal {N}}(0,1)}$

${\displaystyle P_{\vartheta }(B)=P(B-\vartheta )=\int _{B-\vartheta }{\tfrac {1}{\sqrt {2\pi }}}\exp \left(-{\tfrac {1}{2}}x^{2}\right)\mathrm {d} \lambda (x)=\int _{B}{\tfrac {1}{\sqrt {2\pi }}}\exp \left(-{\tfrac {1}{2}}(x-\vartheta )^{2}\right)\mathrm {d} \lambda (x)}$.

So , the location class consists exactly of the normal distributions with variance one and expected value : ${\displaystyle P_{\vartheta }={\mathcal {N}}(\vartheta ,1)}$ ${\displaystyle \vartheta }$

${\displaystyle {\mathcal {P}}_{L}=\{{\mathcal {N}}(\vartheta ,1)\,|\,\vartheta \in \mathbb {R} \}}$.

It should be noted, however, that for all distributions, as in the example above, a shift by on the x-axis does not correspond to a change in the position parameter of the distribution by . An example of this would be the geometric distribution with the expected value as the location parameter. ${\displaystyle \vartheta }$ ${\displaystyle \vartheta }$

properties

A probability measure and the generated location class are given . Then: ${\displaystyle P}$ ${\displaystyle {\mathcal {P}}_{L}}$

• ${\displaystyle {\mathcal {P}}_{L}}$is a dominated distribution class if and only if is dominated, that is, absolutely continuous with respect to a σ-finite measure . ${\displaystyle P}$
• The following applies more strongly: is absolutely continuous with respect to a σ-finite measure if and only if all are absolutely continuous with respect to. ${\displaystyle P}$ ${\displaystyle \lambda }$ ${\displaystyle P_{\vartheta }\in {\mathcal {P}}_{L}}$ ${\displaystyle \lambda }$
• Denoting by generated Lokationsklasse so true . ${\displaystyle {\mathcal {P}}_{L}(P)}$ ${\displaystyle P}$ ${\displaystyle {\mathcal {P}}_{L}(P^{\otimes n})={\mathcal {P}}_{L}(P)^{\otimes n}}$
• Each location class is a Q-invariant distribution class with respect to

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

literature

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