Distribution class
A distribution class or distribution family (also class of distributions or family of distributions ) is understood in stochastics and statistics as a set of probability measures that are characterized by a common, more or less abstract property. The restriction to such properties often makes it possible to show stronger statements with the additionally available structure. An example of this is the Cramer-Rao inequality ; with it the restriction to the exponential family provides a sharp estimate.
term
The term distribution class or distribution family / family of distributions is not used uniformly or in different ways in the literature.
- Some authors use the term for a set of probability measures indexed with an arbitrary index set, i.e. a family in the general mathematical sense .
- Others use it for clearly defined probability distributions, the probability density function or probability function of which is determined via one or more, mostly real parameters, as in the case of the gamma distribution or the binomial distribution .
- A middle way is the definition as a set of probability measures which all have a common property and whose generality is determined by the definition of these properties.
The first meaning is very broad, the second very narrow. Usually the third meaning is used.
Important distribution classes
Some important distribution classes are listed and described below. The definitions of some distribution classes are purely probabilistic , while others are mainly used in mathematical statistics . There are also distribution classes that are used in both subject areas.
- The exponential family : It is characterized by a general density function. It includes the normal distribution , binomial distribution , multinomial distribution , Poisson distribution , gamma distribution and inverse normal distribution . Numerous strong results can be shown in statistics for exponential families.
- The Panjer distribution describes, among other things, the negative binomial distribution , the binomial distribution and the Poisson distribution through a common probability function.
- A dominated distribution class only contains probability measures that are absolutely continuous with respect to another measure. Thus, probability density functions always exist for such classes , which means that the maximum likelihood method can be used, for example .
- A location class results from a given probability distribution by shifting along the x-axis.
- Likewise, a family of scales emerges from a probability distribution through expansion and compression.
- Both location classes and scale families are special cases of Q-invariant distribution classes .
- For distribution classes with monotone density quotient this can be Neyman-Pearson lemma generalize and thus provides far-reaching results in the test theory.
There are also, for example, alpha-stable distributions or infinitely divisible distributions .
literature
- Achim Klenke: Probability Theory . 3. Edition. Springer-Verlag, Berlin Heidelberg 2013, ISBN 978-3-642-36017-6 , doi : 10.1007 / 978-3-642-36018-3 . * Klaus D. Schmidt: Measure and probability . 2nd, revised edition. Springer-Verlag, Heidelberg Dordrecht London New York 2011, ISBN 978-3-642-21025-9 , doi : 10.1007 / 978-3-642-21026-6 .
- Claudia Czado, Thorsten Schmidt: Mathematical Statistics . Springer-Verlag, Berlin Heidelberg 2011, ISBN 978-3-642-17260-1 , doi : 10.1007 / 978-3-642-17261-8 .
- Norbert Kusolitsch: Measure and probability theory . An introduction. 2nd, revised and expanded edition. Springer-Verlag, Berlin Heidelberg 2014, ISBN 978-3-642-45386-1 , doi : 10.1007 / 978-3-642-45387-8 .
- Ludger Rüschendorf: Mathematical Statistics . Springer Verlag, Berlin Heidelberg 2014, ISBN 978-3-642-41996-6 , doi : 10.1007 / 978-3-642-41997-3 .