Maximin test

A Maximin test is a special statistical test in test theory , a branch of mathematical statistics . A Maximin test is a test in which the highest possible probability of an error of the second type is smaller than that of any further test at a given level. The advantage of Maximin tests compared to, for example, equally best tests is that the former already exist under significantly weaker additional assumptions and thus provide a more manageable optimality criterion.

definition

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

Let be the set of all statistical tests on the level . One is called a Maximin test to the level if ${\displaystyle {\mathcal {T}}_{\alpha }}$ ${\displaystyle \alpha }$ ${\displaystyle \Psi \in {\mathcal {T}}_{\alpha }}$ ${\displaystyle \alpha }$

${\displaystyle \inf _{\vartheta \in \Theta _{1}}\operatorname {E} _{\vartheta }\Psi =\sup _{\Phi \in {\mathcal {T}}_{\alpha }}\inf _{\vartheta \in \Theta _{1}}E_{\vartheta }\Phi }$

applies.

interpretation

For fixed , the selectivity corresponds to the test at the point . So is ${\displaystyle \vartheta \in \Theta _{1}}$ ${\displaystyle \operatorname {E} _{\vartheta }\Psi }$ ${\displaystyle \Psi }$ ${\displaystyle \vartheta }$

${\displaystyle \inf _{\vartheta \in \Theta _{1}}\operatorname {E} _{\vartheta }\Psi }$

the lower bound of the selectivity of the test and thus the upper bound for the probability of making an error of the second kind . ${\displaystyle \Psi }$

A Maximin test is therefore a test in which this worst-case probability for an error of the second type is less than or equal to that of any other test.

existence

The existence of Maximin 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 , a maximin test for the level exists. ${\displaystyle \mu }$ ${\displaystyle (P_{\vartheta })_{\vartheta \in \Theta _{0}}}$ ${\displaystyle (P_{\vartheta })_{\vartheta \in \Theta _{1}}}$ ${\displaystyle \alpha }$

literature

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