Consistent test suite

In test theory , a branch of mathematical statistics , a consistent test sequence denotes a sequence of statistical tests that is characterized by the fact that the selectivity of the sequence converges to 1 as the sample size increases. The probability of a type 2 error therefore disappears as the data volume increases. The sequence of tests is usually based on the same idea; the construction as a sequence of tests merely formalizes the ever-increasing sample.

Sometimes there is also the designation as a consistent sequence of tests or simply as a consistent test , the latter being technically incorrect.

definition

A set of probability measures and a disjoint decomposition of into a null hypothesis and an alternative are given . ${\displaystyle (P_{\vartheta })_{\vartheta \in \Theta }}$ ${\displaystyle \Theta }$ ${\displaystyle \Theta _{0}}$ ${\displaystyle \Theta _{1}}$

Let independently and identically distributed random variables be selected according to a distribution . Is a series of statistical tests ${\displaystyle (X_{n})_{n\in \mathbb {N} }}$ ${\displaystyle (P_{\vartheta })_{\vartheta \in \Theta }}$ ${\displaystyle (\Phi _{n})_{n\in \mathbb {N} }}$

${\displaystyle \Phi _{n}\colon (X_{1},\dots ,X_{n})\to [0,1]}$,

so it's called a consistent test suite , if

1. All tests have the same level , i.e. the same bound for an error of type I. With the quality function it is therefore true ${\displaystyle \alpha }$ ${\displaystyle G_{\Phi _{n}}}$

   ${\displaystyle \sup _{\vartheta \in \Theta _{0}}G_{\Phi _{n}}(\vartheta )\leq \alpha \quad \mathrm {f{\ddot {u}}r\;alle\;n} }$

2. The selectivity of the sequence of tests converges to 1. It is therefore true

   ${\displaystyle \lim _{n\to \infty }G_{\Phi _{n}}(\vartheta )=1\quad \mathrm {f{\ddot {u}}r\;alle\;} \vartheta \in \Theta _{1}}$.

comment

The consistency of a test sequence is a purely asymptotic quality feature. Thus, from the consistency it cannot be concluded how fast the selectivity converges, nor from which sample size (for example monotonous) convergence behavior appears. The latter effect is based on the fact that the convergence of a sequence is independent of the first, finitely many, subsequent members.

Even with an infinitely large sample, a consistent test sequence does not always deliver a perfect decision that correctly distinguishes the null hypothesis and the alternative. The probability of an error of the second type disappears, but an error of the first type is still possible in principle.

Web links

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

literature

• Ludger Rüschendorf: Mathematical Statistics . Springer Verlag, Berlin Heidelberg 2014, ISBN 978-3-642-41996-6 , doi : 10.1007 / 978-3-642-41997-3 .