Unadulterated test

An unadulterated test is a special statistical test in test theory , a branch of mathematical statistics . Unadulterated tests are characterized by the fact that the probability of making an error of the first type does not exceed a certain predetermined value, but at the same time they have a lower probability of making an error of the second type than the "trivial test", which is a purely random decision triggers.

definition

A test with a null hypothesis and an alternative is given . Be ${\displaystyle \Phi }$ ${\displaystyle \Theta _{0}}$ ${\displaystyle \Theta _{1}}$

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

the quality function . Then an unadulterated test of the level is called if ${\displaystyle \Phi }$ ${\displaystyle \alpha }$

${\displaystyle G_{\Phi }(\vartheta _{0})\leq \alpha \leq G_{\Phi }(\vartheta _{1})\quad \mathrm {f{\ddot {u}}r\;alle\;} \vartheta _{0}\in \Theta _{0}{\text{ und alle }}\vartheta _{1}\in \Theta _{1}}$

applies. Thus, the unadulterated estimators for the level are precisely the estimators for the level whose discriminatory power is always above the level. ${\displaystyle \alpha }$ ${\displaystyle \alpha }$

use

Unadulterated tests provide a well-founded restriction on the amount of all tests at a given level. The restriction to unadulterated tests thus provides, for example, stronger statements about the existence of consistently best tests .

Similar tests are a central aid for finding consistently best, unadulterated tests .

Asymptotically unadulterated tests

The asymptotically unadulterated tests are a weakening of unadulterated tests. With them, the authenticity only occurs in the limit value with ever larger samples.

Related terms

The corresponding term for confidence ranges in the sense of the duality of tests and confidence ranges are the unadulterated confidence ranges .

Web links

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

literature

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

Individual evidence

1. ↑ OV Shalaevskii: Asymptotically Unbiased-test . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).