Rigorous test

A rigorous test is a special statistical test in test theory , a branch of mathematical statistics . Strict tests, like Maximin tests , gain their importance from the fact that, in contrast to consistently best tests, they already exist under weak conditions.

definition

A (not necessarily parametric) statistical model and a disjoint decomposition of the index set into the null hypothesis and alternative are given . ${\displaystyle (X,{\mathcal {A}},(P_{\vartheta })_{\theta \in \Theta })}$ ${\displaystyle \Theta }$ ${\displaystyle \Theta _{0}}$ ${\displaystyle \Theta _{1}}$

Let be the set of all statistical tests on the level . Let be the quality function of the test and ${\displaystyle {\mathcal {T}}_{\alpha }}$ ${\displaystyle \alpha }$ ${\displaystyle G_{\Phi }}$ ${\displaystyle \Phi }$

${\displaystyle \beta _{{\mathcal {T}}_{\alpha }}^{+}(\vartheta ):=\sup _{\Phi \in {\mathcal {T}}_{\alpha }}G_{\Phi }(\vartheta )}$

the envelope power function ( English envelope power function ) of . ${\displaystyle {\mathcal {T}}_{\alpha }}$

One is called a rigorous test to the level , if ${\displaystyle \Psi \in {\mathcal {T}}_{\alpha }}$ ${\displaystyle \alpha }$

${\displaystyle \sup _{\vartheta \in \Theta _{1}}\left(\beta _{{\mathcal {T}}_{\alpha }}^{+}(\vartheta )-G_{\Psi }(\vartheta )\right)=\inf _{\Phi \in {\mathcal {T}}_{\alpha }}\sup _{\vartheta \in \Theta _{1}}\left(\beta _{{\mathcal {T}}_{\alpha }}^{+}(\vartheta )-G_{\Phi }(\vartheta )\right)}$

Explanation

The envelope power function provides for each parameter , the maximum selectivity of the tests in , if present. Thus the expression is ${\displaystyle \vartheta \in \Theta _{1}}$ ${\displaystyle {\mathcal {T}}_{\alpha }}$ ${\displaystyle \vartheta }$

${\displaystyle \beta _{{\mathcal {T}}_{\alpha }}^{+}(\vartheta )-G_{\Psi }(\vartheta )}$

the deficit of the selectivity in relation to the maximum possible selectivity at the point . Hence is ${\displaystyle \Psi }$ ${\displaystyle \vartheta }$

${\displaystyle \sup _{\vartheta \in \Theta _{1}}\left(\beta _{{\mathcal {T}}_{\alpha }}^{+}(\vartheta )-G_{\Psi }(\vartheta )\right)}$

the maximum deficit of the selectivity of the test . ${\displaystyle \Psi }$

A strict test is therefore a test in which the maximum deviation from the maximum possible selectivity (and thus the enveloping quality function) is smaller than in any other test at a given level.

existence

The existence of strict tests can be shown under rather weak conditions. The central aid for this is the weak convergence and the weak - * - convergence in and . ${\displaystyle L^{1}}$ ${\displaystyle L^{\infty }}$

The central statement is that if a σ-finite measure exists such that or is dominated by this measure , there is a strict test of the level . ${\displaystyle \mu }$ ${\displaystyle (P_{\vartheta })_{\vartheta \in \Theta _{0}}}$ ${\displaystyle (P_{\vartheta })_{\vartheta \in \Theta _{1}}}$ ${\displaystyle \alpha }$

literature

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