Dominated distribution class

A dominant distribution class in mathematical statistics is a set of probability measures that are all absolutely continuous with respect to a measure . Statistical models with dominated distribution classes are easier to handle than those without, since the existence of a probability density is guaranteed and methods such as the maximum likelihood method can be used. In addition, easily manageable criteria for sufficiency and minimal sufficiency exist for dominated distribution classes .

definition

A measurement room and a set of probability measures on this measurement room are given. The set is then called a dominated distribution class if there is a σ-finite measure such that it holds for all that ${\displaystyle (\Omega ,\Sigma )}$ ${\displaystyle {\mathcal {P}}}$ ${\displaystyle {\mathcal {P}}}$ ${\displaystyle \mu }$ ${\displaystyle P\in {\mathcal {P}}}$

${\displaystyle P\ll \mu }$

applies. So each is absolutely continuous with respect to , that means for all with also applies . This is then also noted. ${\displaystyle P}$ ${\displaystyle \mu }$ ${\displaystyle A\in \Sigma }$ ${\displaystyle \mu (A)=0}$ ${\displaystyle P(A)=0}$ ${\displaystyle {\mathcal {P}}\ll \mu }$

Examples

• By definition, the exponential family is a dominated distribution class because it is defined as precisely the distribution class that has a given density with respect to a measure.
• If the distribution class defines precisely those probability measures that have a probability density, then this is also a dominated distribution class. The dominant measure here is the Lebesgue measure . ${\displaystyle {\mathcal {P}}}$ ${\displaystyle \mathbb {R} }$
• If the Cantor distribution is and the above is used to define the new distribution class as , then it is not clear per se whether it is a dominated distribution class or not. is no longer dominated by the Lebesque measure, since the Cantor distribution has no density with respect to the Lebesgue measure. However, it is not clear whether there is another σ-finite measure that dominates, or whether such a measure cannot exist and thus makes the distribution class a non-dominated distribution class. ${\displaystyle C}$ ${\displaystyle {\mathcal {P}}}$ ${\displaystyle {\mathcal {P}}^{*}={\mathcal {P}}\cup C}$ ${\displaystyle {\mathcal {P}}^{*}}$ ${\displaystyle {\mathcal {P}}^{*}}$ ${\displaystyle {\mathcal {P}}^{*}}$

properties

• If a distribution class is dominated, this class is also always dominated by a probability measure. Because if there is a σ-finite measure that dominates the distribution class, then we can get through ${\displaystyle {\mathcal {P}}}$ ${\displaystyle \mu _{\sigma }}$

${\displaystyle \mu _{W}(A):=\sum _{n=1}^{\infty }{\frac {1}{2^{n}}}{\frac {\mu _{\sigma }(A_{n}\cap A)}{\mu _{\sigma }(A_{n})}}}$

define a probability measure that dominates the distribution class. They are a decomposition of with , as is required in the definition of the σ-finite measure. ${\displaystyle A_{n}}$ ${\displaystyle \Omega }$ ${\displaystyle 0<\mu _{\sigma }(A_{n})<\infty }$

• If there is a dominated distribution class, then there always exists such that and is a countable convex combination with really positive coefficients of elements . So it applies ${\displaystyle {\mathcal {P}}}$ ${\displaystyle P^{*}}$ ${\displaystyle {\mathcal {N}}_{\mathcal {P}}={\mathcal {N}}_{P^{*}}}$ ${\displaystyle P^{*}}$ ${\displaystyle {\mathcal {P}}}$

${\displaystyle P^{*}=\sum _{i=1}^{\infty }\alpha _{i}P_{i}{\text{ mit }}\alpha _{i}>0{\text{ und }}\sum _{i=1}^{\infty }\alpha _{i}=1}$.

The set denotes all zero sets. This plays an important role in Halmos-Savage's theorem and some of the results derived from it. ${\displaystyle {\mathcal {N}}_{\mathcal {P}}}$ ${\displaystyle {\mathcal {P}}}$ ${\displaystyle P^{*}}$

• If the distribution class is dominated and if the class is the n-fold product dimensions , then is also dominated. ${\displaystyle {\mathcal {P}}}$ ${\displaystyle {\mathcal {P}}^{\otimes n}}$ ${\displaystyle {\mathcal {P}}^{\otimes n}}$
• Is dominated by and is a measurable function and if all image dimensions are below σ-finite, then the distribution class of image dimensions is also dominated by . ${\displaystyle {\mathcal {P}}}$ ${\displaystyle \mu }$ ${\displaystyle f}$ ${\displaystyle P_{f}}$ ${\displaystyle f}$ ${\displaystyle {\mathcal {P}}_{f}}$ ${\displaystyle \mu _{f}}$
• If the total variation metric is separable , then it is dominated. ${\displaystyle {\mathcal {P}}}$ ${\displaystyle {\mathcal {P}}}$
• If the location class created by is then a dominated distribution class is and only if is dominated. ${\displaystyle {\mathcal {P}}_{L}}$ ${\displaystyle P}$ ${\displaystyle {\mathcal {P}}_{L}}$ ${\displaystyle P}$
• If the σ-algebra of the statistical model is separable and the distribution class is dominant, then the distribution class is separable with regard to the total variation metric. ${\displaystyle {\mathcal {A}}}$

use

According to Radon-Nikodým's theorem, there are always probability densities for dominated distribution classes with regard to the dominant measure. In the case of stochastic models that are equipped with a dominated distribution class, this statement of existence enables the use of methods that are based on probability densities. An example of this is the maximum likelihood method .

In addition, there are criteria for dominated distribution classes that facilitate the checking of the sufficiency of σ-algebras and the sufficiency of statistics . Most of these criteria build on the Halmos-Savage theorem using the measure constructed above . One of these criteria is the Neyman criterion , which provides , for example, the sufficiency of the exponential family . ${\displaystyle P^{*}}$

From the Halmos-Savage theorem it can also be deduced that a minimally sufficient σ-algebra always exists for dominated distribution classes . It is supported by the densities of respect generated. ${\displaystyle P}$ ${\displaystyle P^{*}}$

literature

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