Similar test

A similar test is a special statistical test in test theory , a branch of mathematical statistics . Similar tests are characterized by the fact that their quality function is constant over a given range, in the simplest case an interval in the parameter set. Similar tests are important because, under certain circumstances, equally best similar tests are also equally best unadulterated tests .

definition

A statistical model and a decomposition of into a null hypothesis and an alternative are given . It should also be , ${\displaystyle (X,{\mathcal {A}},(P_{\vartheta })_{\vartheta \in \Theta })}$ ${\displaystyle \Theta }$ ${\displaystyle \Theta _{0}}$ ${\displaystyle \Theta _{1}}$ ${\displaystyle J\subset \Theta }$

${\displaystyle G_{\Phi }(\vartheta ):=\operatorname {E} _{\vartheta }(\Phi )}$

the quality function for the test and . Then a test is called -like on if ${\displaystyle \Phi }$ ${\displaystyle \alpha \in (0,1)}$ ${\displaystyle \Phi }$ ${\displaystyle \alpha }$ ${\displaystyle J}$

${\displaystyle G_{\Phi }(\vartheta )=\alpha \quad \mathrm {f{\ddot {u}}r\;alle\;} \vartheta \in J}$.

comment

Similar tests can also be defined in non-parametric statistical models: The distribution class is then disjointly broken down into the null hypothesis and the alternative . So then and the merit function has probability measures instead of numbers as arguments, so ${\displaystyle {\mathcal {P}}}$ ${\displaystyle {\mathcal {P}}_{0}}$ ${\displaystyle {\mathcal {P}}_{1}}$ ${\displaystyle J\subset {\mathcal {P}}}$

${\displaystyle G_{\Phi }:{\mathcal {P}}\to \mathbb {R} ,\quad G_{\Phi }(P)=\operatorname {E} _{P}(\Phi )}$.

The definition is then analogous, i.e. is -like on if ${\displaystyle \Phi }$ ${\displaystyle \alpha }$ ${\displaystyle J}$

${\displaystyle G_{\Phi }(P)=\alpha \quad \mathrm {f{\ddot {u}}r\;alle\;} P\in J}$.

Properties and use

If you choose or , you can establish relationships with unadulterated tests . Then a topology on or given and a lot of tests and the quality functions of all tests in continuous, then any genuine test of is a -like test . ${\displaystyle {\overline {\Theta _{0}}}\cap {\overline {\Theta _{1}}}=J}$ ${\displaystyle {\overline {{\mathcal {P}}_{0}}}\cap {\overline {{\mathcal {P}}_{1}}}=J}$ ${\displaystyle J\neq \emptyset }$ ${\displaystyle \Theta }$ ${\displaystyle {\mathcal {P}}}$ ${\displaystyle {\mathcal {T}}}$ ${\displaystyle {\mathcal {T}}}$ ${\displaystyle {\mathcal {T}}}$ ${\displaystyle \alpha }$ ${\displaystyle J}$

Conversely, under the same conditions, it is also true that an equally best -like test is also an equally best unadulterated test for the level . ${\displaystyle \alpha }$ ${\displaystyle J}$ ${\displaystyle \alpha }$

Thus, under these conditions, it is sufficient to find consistently best unadulterated tests to find consistently best similar tests. However, these can be characterized by the behavior on the edge between the null hypothesis and the alternative.

Web links

• MS Nikulin: Similar test . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).

literature

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