Randomized test

In test theory , a branch of mathematical statistics, a special class of statistical tests is called randomized tests . In contrast to the non-randomized tests, they do not always make an unambiguous yes / no decision, but instead, if certain data occurs, require a lottery to be carried out to determine the decision.

One of the advantages of randomized tests is that they are easier to handle mathematically. This makes it easier to show optimality properties for randomized tests than for non-randomized tests. An example of this is the Neyman-Pearson test , which is randomized to make the most of its level.

definition

A statistical model is given . A test is a statistic ${\ displaystyle (X, {\ mathcal {A}}, (P _ {\ vartheta}) _ {\ vartheta \ in \ Theta})}$ ${\ displaystyle \ varphi}$

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

,

which assigns a decision to each observation . The decision is coded with 1 = "Rejection of the null hypothesis " and 0 = "Retention of the null hypothesis" ${\ displaystyle x \ in X}$ ${\ displaystyle \ varphi (x)}$

A test is now called randomized if it not only assumes the values 0 and 1, but also values in the interval . The amount ${\ displaystyle \ varphi}$ ${\ displaystyle (0,1)}$

{\ displaystyle \ {x \ in X \ mid \ varphi (x) \ in (0,1) \}}

is then called the randomization range of the test and contains all values for which the test does not make a clear decision.

interpretation

If the decision is coded as above with 1 = "Rejection of the null hypothesis" and 0 = "Retention of the null hypothesis", then the values between 0 and 1 are interpreted as probabilities of making a decision to reject the null hypothesis. A value of the randomized testing at the site , that would therefore that in observation of mean with a probability of 50%, the null hypothesis is rejected and is retained with a probability of 50% of the null hypothesis. To determine the decision, a fair coin would have to be tossed, which then decides whether to reject it or keep it. More generally, it means that there is a probability that the null hypothesis will be rejected and there is a probability that the null hypothesis will be retained. ${\ displaystyle 0 {,} 5}$ ${\ displaystyle x}$ ${\ displaystyle \ varphi (x) = 0 {,} 5}$ ${\ displaystyle x}$ ${\ displaystyle \ varphi (x) = \ kappa}$ ${\ displaystyle \ kappa}$ ${\ displaystyle 1- \ kappa}$

example

Is given as a basic set is 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 not relevant for the definition of a randomized test for the time being. ${\ displaystyle X = \ {0,1,2, \ dots, 10 \}}$ ${\ displaystyle {\ mathcal {A}} = {\ mathcal {P}} (X)}$ ${\ displaystyle n = 10}$ ${\ displaystyle \ vartheta = p \ in (0,1)}$

For example, a randomized test would be given by

{\ displaystyle \ varphi (x) = {\ begin {cases} 0 & {\ text {if}} x \ leq 5 \\ {\ tfrac {2} {3}} & {\ text {if}} x = 6 \\ 1 & {\ text {falls}} x \ geq 7 \ end {cases}}}

.

For values less than or equal to five, the null hypothesis is retained, for values greater than or equal to seven, the null hypothesis is rejected, and if the value is six, the null hypothesis is rejected with a probability of . Randomization range would be the six here, so . ${\ displaystyle {\ tfrac {2} {3}}}$ ${\ displaystyle \ {6 \}}$

If the value six occurs in this test, a fair dice could be thrown. If the number is one, two, three or four, the null hypothesis is rejected, otherwise the null hypothesis is retained.

properties

As already mentioned in the introduction, optimality and existence statements can be derived better for randomized tests. This is mainly because the randomized tests form a convex set . This does not apply to non-randomized tests. For convex sets, there are many far-reaching structural statements about topological properties and the existence of minimal places of functionals . These then enable the corresponding optimality statements to be derived.

supporting documents

Claudia Czado, Thorsten Schmidt: Mathematical Statistics . Springer-Verlag, Berlin Heidelberg 2011, ISBN 978-3-642-17260-1 , p. 148 , doi : 10.1007 / 978-3-642-17261-8 .
Ulrich Krengel : Introduction to probability theory and statistics . For studies, professional practice and teaching. 8th edition. Vieweg, Wiesbaden 2005, ISBN 3-8348-0063-5 , p. 100-104 , doi : 10.1007 / 978-3-663-09885-0 .
David Meintrup, Stefan Schäffler: Stochastics . Theory and applications. Springer-Verlag, Berlin Heidelberg New York 2005, ISBN 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. 265-266 , doi : 10.1515 / 9783110215274 .