Form hypotheses

In mathematical statistics, form hypotheses are an extension of the standard formalism for confidence areas . The form hypotheses determine which values ​​should be included in the confidence range and which should not. This enables the formulation of optimality terms for confidence ranges such as, for example, uniformly best confidence ranges . Using the dual concept of the associated test hypotheses , a relationship between confidence ranges and tests can then be established, with which confidence intervals can be constructed from tests and vice versa.

Form hypotheses

A statistical model is given , where the index set is the probability measure. Furthermore, is a function to be estimated ${\displaystyle (X,{\mathcal {A}},(P_{\vartheta })_{\vartheta \in \Theta })}$ ${\displaystyle \Theta }$

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

given, which in the parametric case is usually referred to as a parameter function and maps it into the decision space. ${\displaystyle \Gamma }$

A family is then designated as a form hypothesis , so that and is as well ${\displaystyle g}$ ${\displaystyle ({\tilde {H}}_{\vartheta },{\tilde {K}}_{\vartheta })_{\vartheta \in \Theta }}$ ${\displaystyle {\tilde {H}}_{\vartheta }\subset \Gamma }$ ${\displaystyle {\tilde {K}}_{\vartheta }\subset \Gamma }$

${\displaystyle {\tilde {H}}_{\vartheta }\cap {\tilde {K}}_{\vartheta }=\emptyset }$ for all ${\displaystyle \vartheta \in \Theta }$

Clearly contains all "correct" values ​​which should be covered as far as possible by the confidence range. Analog contains all "incorrect" values ​​that should not be included in the confidence range if possible. ${\displaystyle {\tilde {H}}_{\vartheta }}$ ${\displaystyle {\tilde {K}}_{\vartheta }}$

example

A simple statistical model is given , where the normal distribution with variance is one. The mean value should be estimated, so is ${\displaystyle (\mathbb {R} ,{\mathcal {B}}(\mathbb {R} ),(P_{\vartheta })_{\vartheta \in \mathbb {R} })}$ ${\displaystyle P_{\vartheta }={\mathcal {N}}(\vartheta ,1)}$

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

Thus the index set and the decision space are both the same, it is . ${\displaystyle \Theta =\Gamma =\mathbb {R} }$

Possible form hypotheses would be

${\displaystyle {\tilde {H}}_{\vartheta }=[\vartheta -2;\vartheta +2]}$as well as for everyone . ${\displaystyle {\tilde {K}}_{\vartheta }=\mathbb {R} \setminus {\tilde {H}}_{\vartheta }}$ ${\displaystyle \vartheta \in \mathbb {R} }$

These state that the area which is symmetrical about the mean value should be covered as far as possible, whereas everything outside should not be covered as far as possible. It should be noted that the definition does not require that the hypotheses disjunct the decision space for each . So in this example it would be quite possible to choose or . ${\displaystyle {\tilde {H}}_{\vartheta }\cup {\tilde {K}}_{\vartheta }=\Gamma }$ ${\displaystyle \vartheta }$ ${\displaystyle {\tilde {K}}_{\vartheta }=(\vartheta +2,+\infty )}$ ${\displaystyle {\tilde {K}}_{\vartheta }=\emptyset }$

Confidence ranges for form hypotheses

definition

Is a confidence range

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

given as well as form hypotheses , then a confidence range for the confidence level is called if the following applies to all : ${\displaystyle ({\tilde {H}}_{\vartheta },{\tilde {K}}_{\vartheta })_{\vartheta \in \Theta }}$ ${\displaystyle C}$ ${\displaystyle g}$ ${\displaystyle 1-\alpha }$ ${\displaystyle \vartheta \in \Theta }$

${\displaystyle P_{\vartheta }(\{\gamma \in C\})\geq 1-\alpha }$for everyone . ${\displaystyle \gamma \in {\tilde {H}}_{\vartheta }}$

example

One chooses as form hypotheses

${\displaystyle {\tilde {H}}_{\vartheta }:=\{g(\vartheta )\}}$,

and arbitrary (but disjoint), this is the confidence range for the form hypotheses ${\displaystyle {\tilde {K}}_{\vartheta }}$

${\displaystyle P_{\vartheta }(\{g(\vartheta )\in C\})\geq 1-\alpha }$ for all ${\displaystyle \vartheta \in \Theta }$

and thus corresponds exactly to the classic formulation of a confidence range for the confidence level . ${\displaystyle 1-\alpha }$

Test hypotheses

Analogous to the form hypotheses, the test hypotheses are defined for given form hypotheses. In contrast to these, they are subsets of the index set instead of the decision space . ${\displaystyle \Theta }$ ${\displaystyle \Gamma }$

definition

Form hypotheses are given . Then means defined by ${\displaystyle ({\tilde {H}}_{\vartheta },{\tilde {K}}_{\vartheta })_{\vartheta \in \Theta }}$ ${\displaystyle (H_{\gamma },K_{\gamma })_{\gamma \in \Gamma }}$

${\displaystyle H_{\gamma }:=\{\vartheta \in \Theta \mid \gamma \in {\tilde {H}}_{\vartheta }\}}$

and

${\displaystyle K_{\gamma }:=\{\vartheta \in \Theta \mid \gamma \in {\tilde {K}}_{\vartheta }\}}$

the test hypotheses for the form hypotheses.

example

If you continue the above example, you get

${\displaystyle H_{\gamma }:=\{\vartheta \in \Theta \mid \gamma \in [\vartheta -2;\vartheta +2]\}=[\vartheta -2;\vartheta +2]}$

and

${\displaystyle K_{\gamma }:=\{\vartheta \in \Theta \mid \gamma \notin {\tilde {H}}_{\vartheta }\}=\mathbb {R} \setminus [\vartheta -2;\vartheta +2]}$

In this example, the form hypotheses and the test hypotheses are identical, even if they are formally defined on different sets: once on the index set and once on the decision space . In general, these two quantities do not match. ${\displaystyle \Theta }$ ${\displaystyle \Gamma }$

literature

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