Rejection area and acceptance area

The rejection area , also called rejection area , rejection area or critical area , is a term in test theory , a branch of mathematical statistics . The rejection area clearly contains all data for which the examined statistical test decides in favor of the alternative or rejects the null hypothesis . Similarly, the amount of all data for which the test accepts the null hypothesis is also referred to as the acceptance range or the non-rejection range.

Formal definition

Given is a statistical model with a decomposition of into the null hypothesis and alternative . ${\ displaystyle (X, {\ mathcal {A}}, (P _ {\ vartheta}) _ {\ vartheta \ in \ Theta})}$ ${\ displaystyle \ Theta}$ ${\ displaystyle \ Theta _ {0}}$ ${\ displaystyle \ Theta _ {1}}$

Is a statistical test

{\ displaystyle \ Phi: (X, {\ mathcal {A}}) \ to ([0,1], {\ mathcal {B}} | _ {[0,1]})}

given, the set is called defined by ${\ displaystyle A \ subset X}$

{\ displaystyle A: = \ {x \ in X \, | \, \ Phi (x) = 1 \}}

the rejection area of the test . The amount ${\ displaystyle \ Phi}$

{\ displaystyle B: = \ {x \ in X \, | \, \ Phi (x) = 0 \}}

is then referred to as the acceptance area.

comment

The rejection area and the acceptance area are subsets of the result space (sample space) of a statistical test . In particular, it is independent of the choice of the null hypothesis and alternative and is only one property of the statistical test as a real-valued function .

Only after defining the null hypothesis and alternative can the important properties be determined via the rejection area (in the case of non-randomized tests ): selectivity , level, type 1 error and type 2 error . The question here is usually to choose the rejection area in such a way that these parameters of the test are within the desired range.

example

A binomial model, i.e. a statistical model, is given

{\ displaystyle X = \ {0,1, \ dots, n \}, \, {\ mathcal {A}} = {\ mathcal {P}} (X)}

and the set of probability measures

{\ displaystyle {\ mathcal {P}} = \ {\ operatorname {Bin} _ {n, \ vartheta} \, | \, \ vartheta \ in [0,1] \}}

.

Then for one with a statistical test is defined by ${\ displaystyle c \ in \ mathbb {N}}$ ${\ displaystyle c <n}$

{\ displaystyle \ Phi (x) = {\ begin {cases} 0 & {\ text {falls}} \ quad x \ leq c \\ 1 & {\ text {falls}} \ quad x> c \ end {cases}} }

.

This test rejects the null hypothesis if there are more than "successes" in the sample. The rejection area of the test is then ${\ displaystyle c}$

{\ displaystyle A = \ {c + 1, \ dots, n \}}

and the assumption range

{\ displaystyle B = \ {1, \ dots, c \}}

.

properties

In the case of one-dimensional questions (e.g. test for the position of a parameter), a distinction is made between one-sided and two-sided tests, depending on the selection of the rejection area. Many questions are multidimensional, for example the question of independence in a larger (more than 2 rows and columns) contingency table . An extreme case are adaptation tests , which are theoretically infinite-dimensional.

In order to facilitate the interpretation of a test, rejection areas are usually selected in a connected manner. In the case of discrete questions (for example Fisher's exact test ), it is possible to better exploit the rejection area by selecting individual cases.

Individual evidence

^ Hans-Otto Georgii: Stochastics . Introduction to probability theory and statistics. 4th edition. Walter de Gruyter, Berlin 2009, ISBN 978-3-11-021526-7 , p. 266 , doi : 10.1515 / 9783110215274 .
↑ Claudia Czado, Thorsten Schmidt: Mathematical Statistics . Springer-Verlag, Berlin Heidelberg 2011, ISBN 978-3-642-17260-1 , p. 158 , doi : 10.1007 / 978-3-642-17261-8 .
↑ Jürgen Hedderich, Lothar Sachs : Applied statistics . Collection of methods with R. 15th edition. Springer Spectrum, Berlin Heidelberg 2016, ISBN 978-3-662-45690-3 , p. 435 , doi : 10.1007 / 978-3-662-45691-0 .

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 .
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 .

[Georgii266-1] Hans-Otto Georgii: Stochastics . Introduction to probability theory and statistics. 4th edition. Walter de Gruyter, Berlin 2009, ISBN 978-3-11-021526-7 , p. 266 , doi : 10.1515 / 9783110215274 .

[Czado158-2] Claudia Czado, Thorsten Schmidt: Mathematical Statistics . Springer-Verlag, Berlin Heidelberg 2011, ISBN 978-3-642-17260-1 , p. 158 , doi : 10.1007 / 978-3-642-17261-8 .

[Hedderich435-3] Jürgen Hedderich, Lothar Sachs : Applied statistics . Collection of methods with R. 15th edition. Springer Spectrum, Berlin Heidelberg 2016, ISBN 978-3-662-45690-3 , p. 435 , doi : 10.1007 / 978-3-662-45691-0 .