Monotonic density quotient

A growing or monotonic density quotient , also called a growing or monotonic likelihood quotient , is a property of a distribution class or a statistical model in mathematical statistics . The Neyman-Pearson lemma can be generalized for models with increasing density quotients and thus provides the existence of uniformly best estimators .

definition

A statistical model is given . Furthermore, the probability density functions exist for all of them . Define ${\displaystyle (X,{\mathcal {A}},(P_{\vartheta })_{\vartheta \in \Theta })}$ ${\displaystyle \Theta \subset \mathbb {R} }$ ${\displaystyle \vartheta \in \Theta }$ ${\displaystyle f_{\vartheta }(x)}$

${\displaystyle R_{\vartheta \colon {\overline {\vartheta }}}:={\frac {f_{\overline {\vartheta }}(x)}{f_{\vartheta }(x)}}}$

the density quotient function.

There is now a statistic for all ${\displaystyle \vartheta <{\overline {\vartheta }}}$

${\displaystyle T\colon X\to \mathbb {R} }$,

so that the density quotient function is a monotonically increasing function in , the statistical model is called a model with increasing density quotient in . ${\displaystyle T}$ ${\displaystyle T}$

So there is a monotonically increasing function so that ${\displaystyle g_{\vartheta \colon {\overline {\vartheta }}}}$

${\displaystyle R_{\vartheta \colon {\overline {\vartheta }}}=g_{\vartheta \colon {\overline {\vartheta }}}\circ T}$

is.

use

In models with a monotonic density quotient, the Neyman-Pearson lemma can be generalized to one-sided tests . One-sided tests are of the form

${\displaystyle \Theta _{0}=\{\vartheta \in \Theta \,|\,\vartheta \leq \vartheta _{0}\}{\text{ und }}\Theta _{1}=\{\vartheta \in \Theta \,|\,\vartheta >\vartheta _{0}\}}$

or vice versa, where and is a predetermined number. In this case, there is a consistently best test at a given level , which can also be specified explicitly. ${\displaystyle \Theta \subset \mathbb {R} }$ ${\displaystyle \vartheta _{0}}$ ${\displaystyle \alpha }$

A large distribution class with a monotonic density quotient is, for example, the one-parameter exponential 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 .
• 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 .
• Hans-Otto Georgii : Stochastics . Introduction to probability theory and statistics. 4th edition. Walter de Gruyter, Berlin 2009, ISBN 978-3-11-021526-7 , doi : 10.1515 / 9783110215274 .