Consistently best confidence range

A consistently best confidence range , also called consistently best range estimation function or evenly discriminating confidence interval , is a special confidence range in mathematical statistics that also fulfills certain optimality properties.

Preliminary remark

Confidence ranges should always achieve two contradicting goals:

• On the one hand, the confidence range should maintain a certain coverage probability (its confidence level). To guarantee this, the confidence range should be as large as possible.
• On the other hand, a confidence range should also be as small as possible in order to be as informative as possible and to enable the most precise possible estimate of the actual parameters.

If the confidence range is chosen too large, the confidence level is maintained, but the confidence range is uninformative. Conversely, if you choose the confidence range too small, the level will not be adhered to, but the confidence range is informative.

Consistently best confidence ranges are now optimal in the sense that it can be shown that they are those with the smallest volume or the shortest length among the confidence ranges at a given confidence level.

definition

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

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

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

Let be the set of all confidence ranges with a confidence level for the form hypotheses . ${\displaystyle {\mathcal {K}}_{1-\alpha }}$ ${\displaystyle 1-\alpha }$ ${\displaystyle ({\tilde {H}}_{\vartheta },{\tilde {K}}_{\vartheta })_{\vartheta \in \Theta }}$

A confidence range

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

For the form hypotheses, a uniformly best confidence range for the confidence level is called if ${\displaystyle 1-\alpha }$

${\displaystyle P_{\vartheta }(\{\gamma \in C^{*}\})=\inf\{P_{\vartheta }(\{\gamma \in C\})\mid C\in {\mathcal {K}}_{1-\alpha }\}}$

applies to everyone and everyone . ${\displaystyle \gamma \in {\tilde {K}}_{\vartheta }}$ ${\displaystyle \vartheta \in \Theta }$

For each , a consistently best confidence range avoids the unwanted values contained in, i.e. better than any other confidence range for the same form hypothesis and at the same confidence level. ${\displaystyle \vartheta }$ ${\displaystyle {\tilde {K}}_{\vartheta }}$

determination

Consistently best confidence ranges can be determined via the duality of confidence ranges and tests . A confidence range is a consistently best confidence range if and only if the corresponding non-randomized test is a consistently best test . This relationship can also be used to construct consistently best confidence ranges from consistently best tests.

literature

• Ludger Rüschendorf: Mathematical Statistics . Springer Verlag, Berlin / Heidelberg 2014, ISBN 978-3-642-41996-6 , pp. 240-247 , doi : 10.1007 / 978-3-642-41997-3 .