Neyman-Pearson test

The Neyman-Pearson test is a special statistical test of central importance in test theory , a branch of mathematical statistics . In the case of application, its prerequisites are usually too restrictive; it acquires its significance through the Neyman-Pearson lemma , which states that the Neyman-Pearson test is a consistently best test . Often, on the basis of this result, an attempt is made to extend this property to larger classes of tests by suitable selection of the framework conditions. An example of this would be models with a monotonic density quotient , for which, under certain circumstances, uniformly best one-sided tests exist.

The test is named after Jerzy Neyman and Egon Pearson .

formulation

Framework

A statistical model is given , where the null hypothesis and the alternative are both simple hypotheses . Thus, both the null hypothesis and the alternative are each given by a probability measure. ${\displaystyle (X,{\mathcal {A}},\{P_{0},P_{1}\})}$ ${\displaystyle P_{0}}$ ${\displaystyle P_{1}}$

Furthermore, the null hypothesis has the probability density function and the alternative has the probability density function . ${\displaystyle f_{0}}$ ${\displaystyle f_{1}}$

Define

${\displaystyle R(x):={\begin{cases}{\frac {f_{1}(x)}{f_{0}(x)}}&{\text{falls}}\quad f_{0}(x)>0\\\infty &{\text{falls}}\quad f_{0}(x)=0\end{cases}}}$.

definition

Under the above conditions, a test is called

${\displaystyle \Phi \colon X\to [0,1]}$

a Neyman-Pearson test on the threshold ${\displaystyle c}$ if

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

is.

properties

The construction of the test can be motivated from the maximum likelihood method, which has proven itself in estimation theory . Instead of selecting the parameter for which the observation is most likely, as in estimation theory, the Neyman-Pearson test accepts or rejects the null hypothesis if the corresponding quotient function falls below or exceeds a certain value. Thus, the Neyman-Pearson test is the simplest possible likelihood ratio test . ${\displaystyle R}$

According to the Neyman-Pearson lemma, there is always a Neyman-Pearson test for the level under the above framework conditions . To construct this one chooses as a - quantile of the distribution . Is then so is ${\displaystyle \alpha }$ ${\displaystyle c}$ ${\displaystyle (1-\alpha )}$ ${\displaystyle P_{0}\circ R^{-1}}$ ${\displaystyle P_{0}(R=c)=0}$

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

a Neyman-Pearson test to the level . If, however , the Neyman-Pearson test is given by the level ${\displaystyle \alpha }$ ${\displaystyle P_{0}(R=c)>0}$ ${\displaystyle \alpha }$

${\displaystyle \Phi (x)={\begin{cases}1&{\text{falls}}\quad R(x)>c\\l&{\text{falls}}\quad R(x)=c\\0&{\text{falls}}\quad R(x)<c\end{cases}}}$,

in which

${\displaystyle l:={\frac {\alpha -P_{0}(R>c)}{P_{0}(R=c)}}}$

is.

According to the Neyman-Pearson lemma, the tests obtained in this way are always equally the best tests for the level for the test problem posed above. ${\displaystyle \alpha }$

According to Stein's lemma, the selectivity of the Neyman-Pearson test converges at an exponential rate with increasing sample size towards 1.Thus, the Neyman-Pearson tests are not only equally better than all other tests, but the probability of an error of type 2 converges to 0 with them at high speed.

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 .