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 .
Let a set of probability measures be on and the image measure of the probability measure under the function .
Then a Q-invariant distribution class is called if for each and every that
is.
Examples
Location classes
If you select the group of translations as a group , so
- ,
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.
Scale families
If you choose , but as a group, the group of multiplications with , that is
- ,
then is for a given probability measure on the set
a Q-invariant distribution class, called the probability of generated scales family .
literature
- Ludger Rüschendorf: Mathematical Statistics . Springer Verlag, Berlin Heidelberg 2014, ISBN 978-3-642-41996-6 , doi : 10.1007 / 978-3-642-41997-3 .