Asymptotic test

An asymptotic test is a special type of statistical test in test theory , a branch of mathematical statistics . Asymptotic tests are constructed for ever larger samples in order to transfer the properties and methods obtained in the limit value to finite sample sizes with a certain approximation error. The tests obtained in this way are then also called approximate tests .

A classic example of approximate tests are the Gaussian tests . In their exact form, they are only designed for normal distribution . With the help of the central limit theorem , the tests can also be extended as an asymptotic test to a large class of distributions. This provides good test methods even for inaccessible distributions with a large sample size.

definition

Framework

A basic space is given , provided with a σ-algebra and a family of probability measures , which is provided with an arbitrary index set . Be the n-fold Cartesian product of , as is the n-fold product σ-algebra by and let the family of n-fold product measures the with itself. ${\displaystyle {\mathcal {X}}}$ ${\displaystyle {\mathcal {A}}}$ ${\displaystyle (P_{\vartheta })_{\vartheta \in \Theta }}$ ${\displaystyle \Theta }$ ${\displaystyle {\mathcal {X}}^{n}}$ ${\displaystyle {\mathcal {X}}}$ ${\displaystyle {\mathcal {A}}^{n}}$ ${\displaystyle {\mathcal {A}}}$ ${\displaystyle (P_{\vartheta }^{n})_{\vartheta \in \Theta }}$ ${\displaystyle P_{\vartheta }}$

Asymptotic test

Under the above conditions, let us have a statistical test for each ${\displaystyle n\in \mathbb {N} }$

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

given. Then the sequence is called an asymptotic test. ${\displaystyle (\varphi _{n})_{n\in \mathbb {N} }}$

An asymptotic test is thus a sequence of tests that are defined for successively larger sample sizes.

Level of an asymptotic test

It denotes the Limes superior and the formation of the expected value with regard to the probability measure . ${\displaystyle \limsup }$ ${\displaystyle \operatorname {E} _{n,\vartheta }}$ ${\displaystyle P_{\vartheta }^{n}}$

Let us break down into the null hypothesis and the alternative . Then the asymptotic test is called an asymptotic test for the level if ${\displaystyle \Theta }$ ${\displaystyle \Theta _{0}}$ ${\displaystyle \Theta _{1}}$ ${\displaystyle (\varphi _{n})_{n\in \mathbb {N} }}$ ${\displaystyle \alpha }$

${\displaystyle \limsup _{n\to \infty }\operatorname {E} _{n,\vartheta }(\varphi _{n})\leq \alpha }$ for all ${\displaystyle \vartheta \in \Theta _{0}}$

applies. In the limit value, the expected value is therefore always below when the null hypothesis is present . However, it does not follow from this that the asymptotic test adheres to the level with a finite sample size. Nor is it stated how quickly it approaches the level. ${\displaystyle \alpha }$

example

A probability distribution is given with an expected value of zero and finite variance . Set and define ${\displaystyle P}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \sigma ^{2}}$ ${\displaystyle \Theta =\mathbb {R} }$

${\displaystyle P_{\vartheta }(A)=P(\vartheta +A)}$

Where is the Minkowski sum . ${\displaystyle \vartheta +A:=\{a+\vartheta \mid a\in A\}}$

Then has the expectation and the variance . If one now considers a sequence of random variables that are independently identical according to distributed random variables , the central limit theorem applies to them . With the abbreviation ${\displaystyle P_{\vartheta }}$ ${\displaystyle \vartheta }$ ${\displaystyle \sigma ^{2}}$ ${\displaystyle P_{\vartheta }}$ ${\displaystyle (X_{n})_{n\in \mathbb {N} }}$

${\displaystyle {\overline {X}}_{n}={\frac {X_{1}+\cdots +X_{n}}{n}}}$

so it applies

${\displaystyle \lim _{n\rightarrow \infty }P_{\vartheta }^{n}\left({\frac {{\overline {X}}_{n}-\vartheta }{\sigma /{\sqrt {n}}}}\leq z\right)=\Phi (z).}$

Here the distribution function is the standard normal distribution . If one now defines the null hypothesis for a fix as ${\displaystyle \Phi }$ ${\displaystyle \vartheta _{0}}$

${\displaystyle H_{0}:=\{\vartheta \in \Theta \mid \vartheta \leq \vartheta _{0}\}}$

and the alternative as

${\displaystyle H_{1}:=\{\vartheta \in \Theta \mid \vartheta >\vartheta _{0}\}}$,

so you can use the test statistics

${\displaystyle T_{n}(X):={\frac {{\overline {X}}_{n}-\vartheta }{\sigma /{\sqrt {n}}}}}$

the tests

${\displaystyle \varphi _{n}:={\begin{cases}1&{\text{ falls }}T_{n}(X)>k\\0&{\text{ falls }}T_{n}(X)\leq k\end{cases}}}$

define for against . It is the critical value that determines the level. The tests then form an asymptotic test, since each test is defined on the successively larger sample area . ${\displaystyle H_{0}}$ ${\displaystyle H_{1}}$ ${\displaystyle k}$ ${\displaystyle \varphi _{n}}$ ${\displaystyle \varphi _{n}}$ ${\displaystyle (\mathbb {R} ^{n},{\mathcal {B}}(\mathbb {R} ^{n}))}$

If you now choose the -quantile of the standard normal distribution as the critical value for all , the asymptotic test has the level . This follows from the fact that according to the central limit theorem, the distribution of the test statistic converges to the standard normal distribution. The value of can be looked up in the quantile table of the standard normal distribution. ${\displaystyle n}$ ${\displaystyle 1-\alpha }$ ${\displaystyle u_{1-\alpha }}$ ${\displaystyle \alpha }$ ${\displaystyle u_{1-\alpha }}$

Individual evidence

1. ↑ ^{a b} Friedrich Liese, Klaus-J. Miescke: Statistical Decision Theory . Estimation, Testing, and Selection. Springer-Verlag, New York 2008, ISBN 978-0-387-73193-3 , pp. 474 , doi : 10.1007 / 978-0-387-73194-0 .
2. ^ Ludger Rüschendorf: Mathematical Statistics . Springer Verlag, Berlin Heidelberg 2014, ISBN 978-3-642-41996-6 , p. 28 , doi : 10.1007 / 978-3-642-41997-3 .
3. ^ Ludger Rüschendorf: Mathematical Statistics . Springer Verlag, Berlin Heidelberg 2014, ISBN 978-3-642-41996-6 , p. vi , doi : 10.1007 / 978-3-642-41997-3 .
4. ↑ Friedrich Liese, Klaus-J. Miescke: Statistical Decision Theory . Estimation, Testing, and Selection. Springer-Verlag, New York 2008, ISBN 978-0-387-73193-3 , pp. 450 , doi : 10.1007 / 978-0-387-73194-0 .