Consistently best test

A consistently best test ( GB test ), consistently strongest test , consistently sharpest test , consistently most powerful test , or briefly best test is a special statistical test in test theory , a branch of mathematical statistics . Uniformly best tests are characterized by the fact that the probability of a type 1 error is always below a specified limit, but at the same time the probability of a type 2 error is smaller than that of any further test that also has the specified limit for the error 1st type complies. The disadvantage of consistently best tests is that, in contrast to other classes of optimal estimators such as strict tests and maximin tests, they only exist under very limited framework conditions.

Sometimes there is also the term UMP test , which is derived from the English of U niform M ost P owerful ( uniformly sharpest or equally powerful ).

definition

A statistical model and a disjoint decomposition of into the null hypothesis and alternative are given . Let be the set of all statistical tests on the level , that is, all statistics ${\displaystyle (X;{\mathcal {A}},(P_{\vartheta })_{\vartheta \in \Theta })}$ ${\displaystyle \Theta }$ ${\displaystyle \Theta _{0}}$ ${\displaystyle \Theta _{1}}$ ${\displaystyle {\mathcal {T}}_{\alpha }}$ ${\displaystyle \alpha }$

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

for the

${\displaystyle \sup _{\vartheta \in \Theta _{0}}\operatorname {E} _{\vartheta }(\Phi )\leq \alpha }$

applies. Be

${\displaystyle G_{\Phi }(\vartheta ):=\operatorname {E} _{\vartheta }(\Phi )}$

the quality function of the test . The test is then called a uniformly best test (or uniformly selectivity test) for the level if the selectivity of is greater than the selectivity of for all others . So it applies ${\displaystyle \Phi }$ ${\displaystyle \Psi \in {\mathcal {T}}_{\alpha }}$ ${\displaystyle \alpha }$ ${\displaystyle \Phi \in {\mathcal {T}}_{\alpha }}$ ${\displaystyle \Psi }$ ${\displaystyle \Phi }$

${\displaystyle G_{\Psi }(\vartheta )\geq G_{\Phi }(\vartheta )\quad \mathrm {f{\ddot {u}}r\;alle\;} \vartheta \in \Theta _{1}}$.

Alternatively, a uniformly most powerful test may whose quality function be defined as the test on the alternative with the envelope power function ( English envelope power function ) of matches. ${\displaystyle {\mathcal {T}}_{\alpha }}$

existence

Generally best tests do not have to exist. The most important tool for the derivation of statements about existence and for the construction of consistently best tests is the Neyman-Pearson lemma , which is also sometimes called the fundamental lemma of mathematical statistics.

Simple hypotheses

For tests with simple hypotheses, i.e. a single-element null hypothesis and a single-element alternative, the Neyman-Pearson lemma provides the existence of a uniformly best test at a given level . This test is the Neyman-Pearson test , a likelihood ratio test . The only additional requirement is the existence of the probability density functions of the null hypothesis and the alternative. ${\displaystyle \alpha }$

After the Lemma of stone which converges selectivity of the Neyman-Pearson tests at an exponential rate with increasing sample size against . ${\displaystyle 1}$

One-sided tests

In one-parameter models with a monotonic density quotient in, there is a uniformly best one-sided test at a given level , i.e. a test for the null hypothesis and an alternative of the form ${\displaystyle T}$ ${\displaystyle \alpha }$

${\displaystyle \Theta _{0}=\{\vartheta \in \Theta \,|\,\vartheta \leq \vartheta _{0}\}\quad {\text{ und }}\quad \Theta _{1}=\{\vartheta \in \Theta \,|\,\vartheta >\vartheta _{0}\}}$

are. Where and is a predetermined number . The test is then given by ${\displaystyle \Theta \subset \mathbb {R} }$ ${\displaystyle \vartheta _{0}}$ ${\displaystyle \Theta }$

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

The selection must be such that the condition is met. Furthermore, the quality function is monotonic. If the null hypothesis and the alternative are interchanged, the less-equal / greater-equal signs are reversed. ${\displaystyle c,l}$ ${\displaystyle \operatorname {E} _{\vartheta _{0}}(\Phi )=\alpha }$

A large distribution class with a monotonic density quotient is the one-parametric exponential family (if the parametric function is monotonic or the family is in natural parameterization). ${\displaystyle \eta }$

The result of the best one-sided tests is derived directly from the Neyman-Pearson lemma: Due to the monotony of the density quotient, the test of against is an equally best test for everyone , thus a consistently best test of against . Since one can show that the quality function is monotonic, the test holds the level for everyone and is therefore an equally best test for the level of against . ${\displaystyle \vartheta _{0}}$ ${\displaystyle \vartheta }$ ${\displaystyle \vartheta \in \Theta _{1}}$ ${\displaystyle \Phi }$ ${\displaystyle \vartheta }$ ${\displaystyle \Theta _{1}}$ ${\displaystyle \vartheta \in \Theta _{0}}$ ${\displaystyle \alpha }$ ${\displaystyle \alpha }$ ${\displaystyle \Theta _{0}}$ ${\displaystyle \Theta _{1}}$

Further statements

Further statements about existence can be obtained, for example, by restricting them to smaller classes of tests such as unadulterated tests , for which statements can be derived, for example, using similar tests .

Related terms

The dual term for confidence ranges (in the sense of the duality of tests and confidence ranges ) for the equally best test is the uniformly best confidence range .

Web links

• MS Nikulin: Uniformly most-powerful test . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).

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 .