Duality of tests and confidence areas

The duality of tests and confidence ranges , also duality of tests and confidence intervals , is in mathematical statistics a connection between confidence ranges and statistical tests , which makes it possible to construct tests from confidence ranges and vice versa. This means that construction methods can also be transferred from one subject area to the other. Furthermore, this duality is used, for example, to describe the optimality properties of confidence areas.

Introductory example

A statistical model and a measuring room are given . A major difference between statistical tests and confidence intervals is that a test takes 0 or 1 as function values ​​or, in the case of a randomized test, values ​​between zero and one. So a test is a picture ${\displaystyle ({\mathcal {X}},{\mathcal {A}},(P_{\vartheta })_{\vartheta \in \Theta })}$ ${\displaystyle (\Gamma ,{\mathcal {A}}_{\Gamma })}$

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

Confidence intervals, on the other hand, take on quantities as values, i.e. they exclude elements , i.e. they are images ${\displaystyle A_{\Gamma }}$

${\displaystyle C\colon {\mathcal {X}}\to {\mathcal {A}}_{\Gamma }}$

with additional measurability properties , see range estimator for details .

Suppose the model is parametric and the parameter is to be estimated. Then is and is the function to be estimated ( parametric function ) ${\displaystyle \Gamma =\Theta }$

${\displaystyle g(\vartheta )=\vartheta }$.

By definition of a confidence interval with a confidence level, the following applies ${\displaystyle C(X)}$ ${\displaystyle 1-\alpha }$

${\displaystyle P_{\vartheta }(\{x\in {\mathcal {X}}\mid \vartheta \in C(X)\})\geq 1-\alpha \quad \mathrm {f{\ddot {u}}r\;alle\;} \vartheta \in \Theta }$.

If we choose a fixed concrete from , so ${\displaystyle \vartheta _{0}}$ ${\displaystyle \Theta }$

${\displaystyle P_{\vartheta _{0}}(\{x\in {\mathcal {X}}\mid \vartheta _{0}\notin C(X)\})\leq \alpha \qquad }$ (1)

and

${\displaystyle P_{\vartheta }(\{x\in {\mathcal {X}}\mid \vartheta \in C(X)\})\geq 1-\alpha \quad \mathrm {f{\ddot {u}}r\;alle\;} \vartheta \in \Theta \setminus \{\vartheta _{0}\}}$.

Now define a statistical test

${\displaystyle T_{C}\colon {\mathcal {X}}\to [0,1]}$

by

${\displaystyle T_{C}(X)=\mathbf {1} _{\{x\in {\mathcal {X}}\mid \vartheta _{0}\notin C(X)\}}(X)}$,

where the indicator function denotes on the set , this is a statistical test of the hypothesis against the alternative . According to equation (1), it maintains the level of significance . ${\displaystyle \mathbf {1} _{A}}$ ${\displaystyle A}$ ${\displaystyle H_{0}=\vartheta _{0}}$ ${\displaystyle H_{1}=\Theta \setminus \{\vartheta _{0}\}}$ ${\displaystyle \alpha }$

As a concrete example, consider the normal distribution model with known variance and unknown expected value , i.e. the statistical model . A right-hand unlimited confidence interval for the unknown expected value at the confidence level is given by ${\displaystyle \sigma _{0}^{2}}$ ${\displaystyle \mu }$ ${\displaystyle (\mathbb {R} ^{n},{\mathcal {B}}(\mathbb {R} ^{n}),({\mathcal {N}}(\mu ,\sigma _{0}^{2}))_{\mu \in \mathbb {R} })}$ ${\displaystyle 1-\alpha }$

${\displaystyle C(X)=\left[{\overline {X}}-{\tfrac {\sigma _{0}}{\sqrt {n}}}u_{1-\alpha };\infty \right)}$.

Here refers to the - quantile of the standard normal distribution, which consists of Quantiltabelle the standard normal can be removed and ${\displaystyle u_{\alpha }}$ ${\displaystyle \alpha }$

${\displaystyle {\overline {X}}={\frac {1}{n}}\sum _{i=1}^{n}X_{i}}$

the sample mean . It follows for a fixed mean ${\displaystyle \mu _{0}\in \mathbb {R} }$

${\displaystyle \mu _{0}\notin C(X)\iff \mu _{0}<{\overline {X}}-{\tfrac {\sigma _{0}}{\sqrt {n}}}u_{1-\alpha }\iff \mu _{0}+{\tfrac {\sigma _{0}}{\sqrt {n}}}u_{1-\alpha }<{\overline {X}}}$.

As a statistical test of the level of against ${\displaystyle \alpha }$ ${\displaystyle H_{0}=\mu _{0}}$ ${\displaystyle H_{1}=\mathbb {R} \setminus \{\mu _{0}\}}$

${\displaystyle T(X,\mu _{0})={\begin{cases}1&{\text{ falls }}\quad \mu _{0}+{\tfrac {\sigma _{0}}{\sqrt {n}}}u_{1-\alpha }<{\overline {X}}\\0&{\text{ sonst.}}\end{cases}}}$

Duality through form hypotheses

More generally, a bijection between the confidence ranges and the non- randomized tests can be established using the concept of shape hypotheses . Form hypotheses and corresponding test hypotheses for a statistical model and a decision space are given . ${\displaystyle ({\tilde {H}}_{\vartheta },{\tilde {K}}_{\vartheta })_{\vartheta \in \Theta }}$ ${\displaystyle (H_{\gamma },K_{\gamma })_{\gamma \in \Gamma }}$ ${\displaystyle ({\mathcal {X}},{\mathcal {A}},(P_{\vartheta })_{\vartheta \in \Theta })}$ ${\displaystyle (\Gamma ,{\mathcal {A}}_{\Gamma })}$

Non-randomized tests from confidence levels

Let be a confidence range for the form hypotheses for the confidence level . Define the amount for each ${\displaystyle C}$ ${\displaystyle ({\tilde {H}}_{\vartheta },{\tilde {K}}_{\vartheta })_{\vartheta \in \Theta }}$ ${\displaystyle 1-\alpha }$ ${\displaystyle \gamma }$

${\displaystyle A_{C,\gamma }:=\{x\in {\mathcal {X}}\mid \gamma \in C(x)\}}$.

Then is for everyone ${\displaystyle \gamma }$

${\displaystyle \varphi _{\gamma }(x)={\begin{cases}1&{\text{falls}}\quad x\notin A_{C,\gamma }\\0&{\text{ falls}}\quad x\in A_{C,\gamma }\end{cases}}}$

a test of the level for the null hypothesis against the alternative . The amount is therefore exactly the acceptance range of the test . ${\displaystyle \alpha }$ ${\displaystyle H_{\gamma }}$ ${\displaystyle K_{\gamma }}$ ${\displaystyle A_{C,\gamma }}$ ${\displaystyle \varphi _{\gamma }}$

Confidence ranges from non-randomized tests

A non-randomized test on the level of the null hypothesis against the alternative with the acceptance range is given for each . So the tests are of the form ${\displaystyle \gamma }$ ${\displaystyle \alpha }$ ${\displaystyle H_{\gamma }}$ ${\displaystyle K_{\gamma }}$ ${\displaystyle A_{\gamma }}$

${\displaystyle \varphi _{\gamma }(x)={\begin{cases}1&{\text{ falls }}\quad x\notin A_{\gamma }\\0&{\text{ falls }}\quad x\in A_{\gamma }\end{cases}}}$.

Then

${\displaystyle C(x):=\{\gamma \in \Gamma \mid x\in A_{\gamma }\}}$

a confidence range for the confidence level for the form hypotheses ${\displaystyle 1-\alpha }$ ${\displaystyle ({\tilde {H}}_{\vartheta },{\tilde {K}}_{\vartheta })_{\vartheta \in \Theta }}$

Correspondence of the optimality terms

The form hypotheses and the corresponding test hypotheses can not only be used to construct tests, but also the optimality statements of tests can be transferred to confidence ranges and vice versa. The following applies:

A confidence range for the form hypotheses and the confidence level is an evenly best confidence range (or an evenly best undistorted confidence range) if, for each of the tests as described above, an evenly best test (or an evenly best undistorted test) for the level for the null hypothesis against the alternative . ${\displaystyle ({\tilde {H}}_{\vartheta },{\tilde {K}}_{\vartheta })_{\vartheta \in \Theta }}$ ${\displaystyle 1-\alpha }$ ${\displaystyle \gamma }$ ${\displaystyle \varphi _{\gamma }}$ ${\displaystyle \alpha }$ ${\displaystyle H_{\gamma }}$ ${\displaystyle K_{\gamma }}$

literature

• Ludger Rüschendorf: Mathematical Statistics . Springer Verlag, Berlin Heidelberg 2014, ISBN 978-3-642-41996-6 , doi : 10.1007 / 978-3-642-41997-3 .
• Claudia Czado, Thorsten Schmidt: Mathematical Statistics . Springer-Verlag, Berlin Heidelberg 2011, ISBN 978-3-642-17260-1 , doi : 10.1007 / 978-3-642-17261-8 .

Individual evidence

1. ^ Ludger Rüschendorf: Mathematical Statistics . Springer Verlag, Berlin Heidelberg 2014, ISBN 978-3-642-41996-6 , p. 240 , doi : 10.1007 / 978-3-642-41997-3 .
2. ↑ Claudia Czado, Thorsten Schmidt: Mathematical Statistics . Springer-Verlag, Berlin Heidelberg 2011, ISBN 978-3-642-17260-1 , p. 158 , doi : 10.1007 / 978-3-642-17261-8 .