General test

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This article deals with decision-making processes in the field of mathematical statistics. For further meanings, see Decision Making Process (disambiguation) .

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A general test or decision-making process is an abstract instrument of mathematical statistics . Almost all statistical tests, such as hypothesis tests or parameter point estimates , can be mathematically recorded in the form of a general test. The aim of a general test is, on the basis of the (observed) realization of one or more previously defined random variables , whose exact probability distribution i. d. As a rule, it is not known to make a decision with regard to an issue under consideration.

Example: A pharmaceutical company wants to test a newly developed drug for its (unknown) effectiveness. A certain number of patients are given the drug for this purpose. Based on the measured effect of the drug on the patient, the pharmaceutical company now has to decide whether to launch the new drug on the market or whether to continue to use a tried and tested drug.

If the pharmaceutical company decides to launch the new drug, there is a risk that the decision-making process used will only incorrectly classify it as better than the old drug. In this case, the pharmaceutical company would suffer unnecessary damage. In order to avoid this, every general test is based on a so-called damage function, with the help of which one tries to minimize the risk of a decision by choosing a "suitable" decision function .

definition

Given a measuring room and a family of probability measures on . includes all possible realizations or observations. Next is a lot of possible decisions. ${\displaystyle (\Omega ,{\mathcal {A}})}$ ${\displaystyle {\mathcal {F}}=\{P_{\theta }\mid \theta \in \Theta \}}$ ${\displaystyle {\mathcal {A}}}$ ${\displaystyle \Omega }$ ${\displaystyle {\mathcal {D}}}$

• A mapping is called the damage function . ${\displaystyle s:{\mathcal {D}}\times \Theta \to \mathbb {R} _{+}}$

• A mapping is called a general test , decision function or decision procedure if and only if the mapping is just - measurable for each . Here denotes the Borel σ-algebra over . ${\displaystyle \delta :\Omega \to {\mathcal {D}}}$ ${\displaystyle \theta \in \Theta }$ ${\displaystyle \omega \mapsto s(\delta (\omega ),\theta )}$ ${\displaystyle ({\mathcal {A}},{\mathfrak {B}})}$ ${\displaystyle {\mathfrak {B}}}$ ${\displaystyle \mathbb {R} }$

Quality criteria

risk

Let it be a class of decision functions. For an element one denotes ${\displaystyle {\mathcal {T}}}$ ${\displaystyle \delta \in {\mathcal {T}}}$

${\displaystyle r_{\delta }:\Theta \to \mathbb {R} _{+}}$ fortune ${\displaystyle r_{\delta }(\theta ):=\int _{\Omega }s(\delta (\omega ),\theta )\ dP_{\theta }(\omega )}$

as a risk function . This indicates the average damage caused by the application of the test under the distribution . Because of this, this always exists, but possibly improperly. Next is called ${\displaystyle \delta }$ ${\displaystyle P_{\theta }}$ ${\displaystyle s\geq 0}$

${\displaystyle r(\delta ):=\sup _{\theta \in \Theta }r_{\delta }(\theta )}$

than the risk of . ${\displaystyle \delta }$

It has now further an algebra over and a probability at given defined as an a priori distribution , or (subjective) prior distribution on the set of parameters. If the risk function is measurable with respect to , then the so-called Bayesian risk of the test with respect to it can be introduced, and one then sets ${\displaystyle \sigma }$ ${\displaystyle {\mathcal {S}}}$ ${\displaystyle \Theta }$ ${\displaystyle \mu }$ ${\displaystyle (\Theta ,{\mathcal {S}})}$ ${\displaystyle \mu }$ ${\displaystyle \theta \mapsto r_{\delta }(\theta )}$ ${\displaystyle {\mathcal {S}}}$ ${\displaystyle \delta }$ ${\displaystyle \mu }$

${\displaystyle r_{\delta }(\mu ):=\int _{\Theta }r_{\delta }(\theta )\ d\mu (\theta )}$.

Efficiency

With the help of the risk and the risk function, two general tests can now be compared with one another. It is said is at least as efficient as when ${\displaystyle \delta _{1},\delta _{2}\in {\mathcal {T}}}$ ${\displaystyle \delta _{1}}$ ${\displaystyle \delta _{2}}$

${\displaystyle r_{\delta _{1}}(\theta )\leq r_{\delta _{2}}(\theta )\quad \forall \theta \in \Theta }$.

In the case of a preliminary assessment, the tests can also be compared using the Bayesian risk. One then says is at least as efficient as when . ${\displaystyle \mu }$ ${\displaystyle \delta _{1}}$ ${\displaystyle \delta _{2}}$ ${\displaystyle r_{\delta _{1}}(\mu )\leq r_{\delta _{2}}(\mu )}$

Optimality

The optimality of a test can be introduced in many different ways. A test is called a ${\displaystyle \delta ^{*}\in {\mathcal {T}}}$

• highly efficient in , if . ${\displaystyle {\mathcal {T}}}$ ${\displaystyle r_{\delta ^{*}}(\theta )=\min _{\delta \in {\mathcal {T}}}r_{\delta }(\theta )\ \forall \theta \in \Theta }$
• Minimax procedure in , if applies. ${\displaystyle {\mathcal {T}}}$ ${\displaystyle r(\delta ^{*})=\min _{\delta \in {\mathcal {T}}}r(\delta )}$
• Bayesian solution in re . , If applies. ${\displaystyle {\mathcal {T}}}$ ${\displaystyle \mu }$ ${\displaystyle r_{\delta ^{*}}(\mu )=\min _{\delta \in {\mathcal {T}}}r_{\delta }(\mu )}$
• multisubjective optimal or -minimax method in , if a family of probability measures is on and holds . ${\displaystyle {\mathfrak {M}}}$ ${\displaystyle {\mathcal {T}}}$ ${\displaystyle {\mathfrak {M}}}$ ${\displaystyle {\mathcal {S}}}$ ${\displaystyle \sup _{\mu \in {\mathfrak {M}}}r_{\delta ^{*}}(\mu )=\min _{\delta \in {\mathcal {T}}}\sup _{\mu \in {\mathfrak {M}}}r_{\delta }(\mu )}$

With a fixed parameter , the unavoidable damage for each test is in . For a good test you will therefore require that ${\displaystyle \theta }$ ${\displaystyle \inf _{\delta \in {\mathcal {T}}}r_{\delta }(\theta )}$ ${\displaystyle {\mathcal {T}}}$

${\displaystyle \rho (\delta ^{*}):=\sup _{\theta \in \Theta }\left(r_{\delta ^{*}}(\theta )-\inf _{\delta _{\in }{\mathcal {T}}}r_{\delta }(\theta )\right)}$

becomes as small as possible ("minimal regret"). Therefore it is further referred to as ${\displaystyle \delta ^{*}}$

• strictest test in when applies. ${\displaystyle {\mathcal {T}}}$ ${\displaystyle \rho (\delta ^{*})=\min _{\delta \in {\mathcal {T}}}\rho (\delta )}$

Correlation : In the case of the optimality criteria listed here, the highest efficiency can be classified as the strongest requirement, because if a test is extremely efficient in , it is already the minimax method, Bayesian solution, multisubjectively optimal and also the strictest test. ${\displaystyle \delta ^{*}}$ ${\displaystyle {\mathcal {T}}}$

Examples

Hypothesis test

In a hypothesis or significance test , one looks at two mutually exclusive hypotheses and one of which, as a rule, one tries , for example , to reject on the basis of an observation . The set of possible decisions is therefore of the form , where one defines: ${\displaystyle H_{0}}$ ${\displaystyle H_{1}}$ ${\displaystyle H_{0}}$ ${\displaystyle \omega \in \Omega }$ ${\displaystyle {\mathcal {D}}=\{d_{1},d_{2}\}}$

${\displaystyle d_{1}:=}$"Hypothesis can be rejected." ${\displaystyle H_{0}}$

${\displaystyle d_{2}:=}$"The hypothesis cannot be rejected, so no conclusion can be drawn from the experiment." ${\displaystyle H_{0}}$

Parameter point estimation

Given a random variable with respect to two measurement spaces and , which is subject to the distribution family . The "true" parameter is unknown here . This, or more generally a value that depends on , is to be estimated. That is why one regards as a decision-making space . Often used as a damage function ${\displaystyle X:\Omega '\to \Omega }$ ${\displaystyle (\Omega ',{\mathcal {A}}')}$ ${\displaystyle (\Omega ,{\mathcal {A}})}$ ${\displaystyle {\mathcal {F}}=\{P_{\theta }\mid \theta \in \Theta \}}$ ${\displaystyle \theta }$ ${\displaystyle \theta }$ ${\displaystyle \lambda (\theta )}$ ${\displaystyle {\mathcal {D}}=\lambda (\Omega )}$

${\displaystyle s:{\mathcal {D}}\times \Theta \to \mathbb {R} _{+},\ s(d,\theta )=(d-\lambda (\theta ))^{2}}$.

This results in the mean square deviation of the estimate from the value to be estimated for a test as a risk function, because ${\displaystyle \delta :\Omega \to {\mathcal {D}}}$

${\displaystyle r_{\delta }(\theta )=\int _{\Omega }s(\delta (\omega ),\theta )\ dP_{\theta }(\omega )=\mathbb {E} _{\theta }((\delta (X)-\lambda (\theta ))^{2})}$.

Parameter range estimation

The random variable is considered again . One would like to estimate an area in which one suspects the "true" parameter . One sets for this . The empty set is excluded as a decision, since estimating it would not make sense. As a payload, the mapping provides with on. With it you get the risk function for a test ${\displaystyle X}$ ${\displaystyle \theta }$ ${\displaystyle {\mathcal {D}}:={\mathfrak {P}}(\Theta )\backslash \{\emptyset \}}$ ${\displaystyle s:{\mathcal {D}}\times \Theta \to \mathbb {R} _{+}}$ ${\displaystyle s(d,\theta ):=1_{d^{\complement }}(\theta )}$ ${\displaystyle \delta :\Omega \to {\mathcal {D}}}$

${\displaystyle r_{\delta }(\theta )=\int _{\Omega }s(\delta (\omega ),\theta )\ dP_{\theta }(\omega )=\int _{\Omega }1_{\delta (\omega )^{\complement }}(\theta )\ dP_{\theta }(\omega )=P_{\theta }(\{\omega \mid \theta \notin \delta (\omega )\})\ ,}$

d. H. is precisely the probability with which the parameter is not in the estimated set. The error probability of the procedure for the parameter is therefore also called . The risk is called the significance limit of . ${\displaystyle r_{\delta }(\theta )}$ ${\displaystyle \theta }$ ${\displaystyle r_{\delta }(\theta )}$ ${\displaystyle \delta }$ ${\displaystyle \theta }$ ${\displaystyle r(\delta )=\sup _{\theta \in \Theta }r_{\delta }(\theta )}$ ${\displaystyle \delta }$