Quality function
A power function , also selectivity function , power function , test strength function or test sharpness function , a special real-valued function in the test theory , a branch of mathematical statistics . A merit function can be assigned to each statistical test . In the parametric case, this assigns the average decision to each parameter that the test makes when the parameter is actually present. Many statistical concepts such as selectivity or the level of a test can be found in the quality function or can be defined using it.
definition
A (not necessarily parametric) statistical model and a disjoint decomposition of the index set into the null hypothesis and alternative are given . Furthermore, be a statistical test
given. Then the function is called
defined by
the quality function of the test . Here denotes the expected value with regard to the probability measure . Thus, the quality function indicates the probability with which the test will decide in favor of the alternative if the probability measure is present.
Derivable terms
The following terms can be defined using the quality function or can be found in it.
Level of a test
If a test is for the level , then the following applies
- .
The level of a test is thus an upper limit for the quality function of the test and thus also an upper limit for errors of the first type of test. Correspondingly, tests with an effective level are those for which a smallest upper bound for their quality function is on and thus the probability of an error is type I.
Selectivity of a test
The discriminatory power of a test indicates how high the probability is of deciding on the alternative if it really is. Thus, the selectivity of the test for the present is given as . Accordingly, the probability of is type 2 error when there are given by
- .
Enveloping merit function
If a set of tests is given, it is called the function
defined by
the envelope power function (engl. envelope power function ). It assigns the greatest selectivity value of all tests of the set to each element of the alternative . It is used, for example, in the formulation of optimality criteria for tests such as consistently best tests or strict tests . Thus, the consistently best tests are precisely those tests whose quality function on the alternative corresponds to the enveloping quality function.
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 .
- 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 .