Non-randomized test

As a non-randomized test is in the test theory a branch of, mathematical statistics , a special class of statistical tests called. In contrast to the randomized tests , they always provide a clear decision as to whether the hypothesis should be rejected or retained. The main advantage of non-randomized tests is that they are more accessible and understandable. However, they are disadvantageous from a mathematical point of view because they lack some desirable properties. Many statements about existence and optimality cannot be shown for non-randomized tests.

definition

A non-randomized test is clearly an illustration that assigns to each observation whether the null hypothesis is retained or whether it is rejected if this observation is present. The decision is coded with 1 = "Rejection of the null hypothesis" and 0 = "Retention of the null hypothesis" ${\displaystyle \varphi }$ ${\displaystyle x}$

A non-randomized test is strictly formally defined as follows: A statistical model is given . A non-randomized test is a statistic ${\displaystyle (X,{\mathcal {A}},(P_{\vartheta })_{\vartheta \in \Theta })}$

${\displaystyle \varphi \colon (X,{\mathcal {A}})\to ([0,1],{\mathcal {B}}([0,1]))}$,

which only takes the values ​​zero or one. So it is for everyone . ${\displaystyle \varphi (x)\in \{0,1\}}$ ${\displaystyle x\in X}$

example

Consider the basic amount , provided with the power set as σ-algebra , that is . This set can be provided, for example, with the binomial distribution with and as a probability distribution. This and the exact choice of hypotheses are initially irrelevant for the definition of a non-randomized test. ${\displaystyle X=\{0,1,2,\dots ,10\}}$ ${\displaystyle {\mathcal {A}}={\mathcal {P}}(X)}$ ${\displaystyle n=10}$ ${\displaystyle \vartheta =p\in (0,1)}$

For example, a non-randomized test would be given by

${\displaystyle \varphi (x)={\begin{cases}0&{\text{ falls }}x\leq 7\\1&{\text{ falls }}x\geq 8\end{cases}}}$.

The test thus retains the null hypothesis for values ​​less than or equal to seven and rejects it for values ​​greater than or equal to eight. It only takes the values ​​zero and one and is therefore a non-randomized test.

Differentiation from the randomized tests

The main difference between randomized tests and non-randomized tests is that randomized tests also take values ​​between zero and one, i.e. from the interval . These values ​​are then interpreted as the probability of deciding against the null hypothesis. If such a value is available, a corresponding lottery procedure would have to be carried out in order to make the decision. In contrast, non-randomized tests are deterministic in the sense that they always make the same decision when the same data is available. This is not the case with randomized tests. ${\displaystyle (0,1)}$

literature

• Ludger Rüschendorf: Mathematical Statistics . Springer Verlag, Berlin Heidelberg 2014, ISBN 978-3-642-41996-6 , p. 21 , doi : 10.1007 / 978-3-642-41997-3 .
• David Meintrup, Stefan Schäffler: Stochastics . Theory and applications. Springer-Verlag, Berlin Heidelberg New York 2005, ISBN 978-3-540-21676-6 , pp. 483-484 , doi : 10.1007 / b137972 .
• 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 .