Unadulterated confidence range

An unadulterated confidence range , also an unadulterated range estimator or an undistorted confidence range is a special confidence range in mathematical statistics . The authenticity itself is not a concept of optimality, but enables the construction of optimal confidence ranges as well as confidence ranges with minimal volume. If the confidence range is one-dimensional, one speaks accordingly of an undistorted / undistorted confidence interval or of an undistorted interval estimation function .

definition

A statistical model as well as a decision space and a function to be estimated are given ${\displaystyle (X,{\mathcal {A}},(P_{\vartheta })_{\vartheta \in \Theta })}$ ${\displaystyle (\Gamma ,{\mathcal {A}}_{\Gamma })}$

${\displaystyle g\colon \Theta \to \Gamma }$,

which in the parametric case is also referred to as a parameter function.

A confidence range

${\displaystyle C\colon X\to {\mathcal {P}}(\Gamma )}$

for the confidence level is called an unadulterated confidence range, if for all ${\displaystyle 1-\alpha }$ ${\displaystyle \vartheta ,\vartheta '\in \Theta }$

${\displaystyle P_{\vartheta }(\{g(\vartheta )\in C\})\geq P_{\vartheta }(\{g(\vartheta ')\in C\})}$

applies. For each , the probability of covering the correct parameter is greater than the probability of covering any other parameter . ${\displaystyle \vartheta }$ ${\displaystyle g(\vartheta )}$ ${\displaystyle g(\vartheta ')}$

example

Given is the normal distribution model with known variance and unknown expected value ${\displaystyle \sigma _{0}^{2}}$ , i.e. the statistical model , provided with the distribution class . The expected value should be covered , the parameter function is therefore given by ${\displaystyle (\mathbb {R} ^{n},{\mathcal {B}}(\mathbb {R} ^{n}),{\mathcal {P}})}$ ${\displaystyle {\mathcal {P}}=\{{\mathcal {N}}^{\otimes n}(\mu ,{\sigma _{0}}^{2})\;|\;\mu \in \mathbb {R} \}}$ ${\displaystyle \mu }$

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

A mutual confidence range for the expected value is given by, for example ${\displaystyle \mu }$

${\displaystyle C(x)=\left[{\overline {x}}-{\tfrac {\sigma _{0}}{\sqrt {n}}}u_{1-\alpha /2};{\overline {x}}+{\tfrac {\sigma _{0}}{\sqrt {n}}}u_{1-\alpha /2}\right]}$.

The - is the quantile of the standard normal distribution and the sample mean . ${\displaystyle u_{\alpha }}$ ${\displaystyle \alpha }$ ${\displaystyle {\overline {x}}}$

The confidence range is unadulterated because it is forever ${\displaystyle \mu \neq \mu '}$

${\displaystyle {\begin{aligned}P_{\mu }\left(\{\mu '\in C\}\right)&=P_{\mu }\left({\overline {x}}-{\tfrac {\sigma _{0}}{\sqrt {n}}}u_{1-\alpha /2}\leq \mu '\leq {\overline {x}}+{\tfrac {\sigma _{0}}{\sqrt {n}}}u_{1-\alpha /2}\right)\\&=P_{\mu }\left({\frac {\mu '-\mu }{\sigma _{0}/{\sqrt {n}}}}-u_{1-\alpha /2}\leq {\frac {{\overline {x}}-\mu }{\sigma _{0}/{\sqrt {n}}}}\leq {\frac {\mu '-\mu }{\sigma _{0}/{\sqrt {n}}}}+u_{1-\alpha /2}\right)\\&=\Phi \left({\frac {\mu '-\mu }{\sigma _{0}/{\sqrt {n}}}}+u_{1-\alpha /2}\right)-\Phi \left({\frac {\mu '-\mu }{\sigma _{0}/{\sqrt {n}}}}-u_{1-\alpha /2}\right)\end{aligned}}}$,

where denotes the distribution function of the standard normal distribution. But the last expression is maximal for , so the confidence range is unadulterated. ${\displaystyle \Phi (\cdot )}$ ${\displaystyle \mu =\mu '}$

General definition via form hypotheses

Under the same general conditions as above, a confidence range for the form hypotheses and for the confidence level is called an unadulterated confidence range, if for all ${\displaystyle C}$ ${\displaystyle ({\tilde {H}}_{\vartheta },{\tilde {K}}_{\vartheta })_{\vartheta \in \Theta }}$ ${\displaystyle 1-\alpha }$ ${\displaystyle \vartheta \in \Theta }$

${\displaystyle P_{\vartheta }(\{\gamma \in C\})\leq 1-\alpha }$ for all ${\displaystyle \gamma \in {\tilde {K}}_{\vartheta }}$

is.

Each value from the "set to be avoided" is therefore covered less often than each value from the "set to be covered" (see shape hypotheses # confidence ranges for shape hypotheses ) ${\displaystyle {\tilde {K}}_{\vartheta }}$ ${\displaystyle {\tilde {H}}_{\vartheta }}$

The first formulation results from using the form hypotheses

${\displaystyle {\tilde {H}}_{\vartheta }=\{g(\vartheta )\}}$ and ${\displaystyle {\tilde {K}}_{\vartheta }=\{g(\vartheta ')\mid \vartheta '\neq \vartheta \}}$

and assuming that is injective . ${\displaystyle g}$

Related terms

The corresponding term for statistical tests in the sense of the duality of tests and confidence ranges are the unadulterated tests .

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. ↑ Claudia Czado, Thorsten Schmidt: Mathematical Statistics . Springer-Verlag, Berlin Heidelberg 2011, ISBN 978-3-642-17260-1 , p. 142 , doi : 10.1007 / 978-3-642-17261-8 .
2. ^ Ludger Rüschendorf: Mathematical Statistics . Springer Verlag, Berlin Heidelberg 2014, ISBN 978-3-642-41996-6 , p. 241 , doi : 10.1007 / 978-3-642-41997-3 .