Q-invariant distribution class

A Q-invariant distribution class is a special distribution class in mathematical statistics , which is characterized by the fact that the probability measures contained in it are closed with regard to the formation of certain image measures . The special case of a Q-invariant distribution class are the location classes and the scale families .

Q-invariant distribution classes are used, for example, in the investigation of equivariant estimators .

definition

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

Let a set of probability measures be on and the image measure of the probability measure under the function . ${\displaystyle {\mathcal {P}}_{\mathcal {Q}}}$ ${\displaystyle (X,{\mathcal {A}})}$ ${\displaystyle P^{f}}$ ${\displaystyle P}$ ${\displaystyle f}$

Then a Q-invariant distribution class is called if for each and every that ${\displaystyle {\mathcal {P}}_{\mathcal {Q}}}$ ${\displaystyle P\in {\mathcal {P}}_{\mathcal {Q}}}$ ${\displaystyle q\in {\mathcal {Q}}}$

${\displaystyle P^{q}\in {\mathcal {P}}_{\mathcal {Q}}}$

is.

Examples

Location classes

If you select the group of translations as a group , so ${\displaystyle (X,{\mathcal {A}})=(\mathbb {R} ,{\mathcal {B}}(\mathbb {R} ))}$ ${\displaystyle \mathbb {R} }$

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

a location class would be a Q-invariant distribution class, because the location classes arise precisely from the shift of a probability measure along the x-axis.

Conversely, however, not every Q-invariant distribution class with the one defined above is a location class. The Q-invariant distribution class could, for example, have arisen from two or more different probability distributions by displacement, which is not possible with location classes, because these are always displacements of a measure. Associations of Q-invariant distribution classes are obviously Q-invariant again, but this does not apply to location classes. ${\displaystyle {\mathcal {Q}}}$

Scale families

If you choose , but as a group, the group of multiplications with , that is ${\displaystyle (X,{\mathcal {A}})=(\mathbb {R} _{+}^{n},{\mathcal {B}}(\mathbb {R} _{+}^{n}))}$ ${\displaystyle \vartheta \in (0,\infty )}$

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

then is for a given probability measure on the set ${\displaystyle P}$ ${\displaystyle (\mathbb {R} _{+}^{n},{\mathcal {B}}(\mathbb {R} _{+}^{n}))}$

${\displaystyle {\mathcal {P}}_{\mathcal {Q}}:=\{P^{S}\,|\,S\in {\mathcal {Q}}\}}$

a Q-invariant distribution class, called the probability of generated scales family . ${\displaystyle P}$

literature

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