Risk function

Risk function is a term from mathematical statistics and is used there in the context of general statistical decision problems . The risk function indicates how great the "damage" to be expected when using a given decision function . Risk functions play a role in the determination of optimal decision-making functions, since risk functions can be used to define an order relationship between the decision-making functions. This makes it possible to search for optimal elements among subsets of the decision functions.

definition

Given is a statistical decision problem , i.e. a statistical model , a decision space and a loss function . Furthermore, let the set of randomized decision functions be and . Then the function is called ${\displaystyle ({\mathcal {E}},\Omega ,L)}$ ${\displaystyle {\mathcal {E}}=(X,{\mathcal {A}},(P_{\vartheta })_{\vartheta \in \Theta })}$ ${\displaystyle (\Omega ,\Sigma )}$ ${\displaystyle L:\Theta \times \Omega \to {\overline {\mathbb {R} }}_{+}}$ ${\displaystyle {\mathcal {D}}}$ ${\displaystyle \delta \in {\mathcal {D}}}$

${\displaystyle R:\Theta \times {\mathcal {D}}\to \mathbb {R} _{+}}$

defined by

${\displaystyle R(\vartheta ,\delta ):=\int _{X}\int _{\Omega }L(\vartheta ,y)\delta (x,\mathrm {d} y)\mathrm {d} P_{\vartheta }}$

a risk function. It indicates how large the expected “loss” is when using the decision function , if the parameter is present. ${\displaystyle \delta }$ ${\displaystyle \vartheta }$

If you consider the risk function as a function in for the fixed , you also write . The set of these risk functions is then defined as and this set is called the risk set. ${\displaystyle \vartheta }$ ${\displaystyle \delta }$ ${\displaystyle R_{\delta }(\vartheta )}$ ${\displaystyle {\mathcal {R}}:=\{R_{\delta }\,|\,\delta \in {\mathcal {D}}\}}$

Risk functions of non-random decision functions

If a non- randomized decision function and the corresponding representation is a randomized decision function, where the Dirac measure is referred to here, then the result is a risk function ${\displaystyle d}$ ${\displaystyle \delta _{d}(x,A)=\Delta _{\{d(x)\}}(A)}$ ${\displaystyle \Delta }$

${\displaystyle R_{\delta }(\vartheta )=\int _{X}\int _{\Omega }L(\vartheta ,y)\mathrm {d} \Delta _{\{d(x)\}}\mathrm {d} P_{\vartheta }=\int _{X}L(\vartheta ,d(x))\mathrm {d} P_{\vartheta }(x)=\operatorname {E} _{\vartheta }L(\vartheta ,d)}$,

thus the expected value of the loss.

example

If one uses the Gaussian loss in estimation theory

${\displaystyle L_{2}(\vartheta ,\cdot ):=\left(\cdot -g(\vartheta )\right)^{2}}$

for the evaluation of real-valued point estimators , the mean square error is obtained as the risk function

${\displaystyle R(\vartheta ,T)=\operatorname {MSE} (T,\vartheta ):=\operatorname {E} _{\vartheta }\left(\left(T-g(\vartheta )\right)^{2}\right)}$.

When restricting to unbiased estimators, the risk function is then reduced to the variance of the estimator, i.e.

${\displaystyle R(\vartheta ,T)=\operatorname {Var} _{\vartheta }(T)}$.

Similarly, when using the Laplace loss, the mean error in terms of amount is obtained as a risk function.

Remarks

Game theory interpretation

Finding an optimal decision function can be viewed as a game in the game theory sense . First nature selects a parameter as a pure strategy from the set of strategies , the statistician then answers with the choice of a mixed strategy that corresponds to the choice of a decision function from the set of strategies . The risk function is then the payout function of this two-person zero-sum game . ${\displaystyle \vartheta }$ ${\displaystyle \Theta }$ ${\displaystyle {\mathcal {D}}}$

Equalizer

A decision function for which the risk function is constant, that is ${\displaystyle \delta _{0}}$ ${\displaystyle \vartheta }$

${\displaystyle R(\vartheta ,\delta _{0})=c\quad \mathrm {f{\ddot {u}}r\;alle\;} \vartheta \in \Theta }$

for a valid one is Egalisator ( English equalizer rule ). These play a role in the relationships between the various optimality criteria for decision-making functions. ${\displaystyle c\in \mathbb {R} }$

Generalizations

A generalization of the risk function is the Bayesian risk . Here one does not consider the evaluation of individual , but considers probability measures on the so-called a priori distributions . These can be interpreted as prior information about the distribution of the parameter. From the game theory perspective, Bayesian risk is the payoff function of the mixed extension of the game described above. ${\displaystyle \vartheta \in \Theta }$ ${\displaystyle \Theta }$

literature

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

Individual evidence

1. ↑ G. Bamberg: Statistical Decision Theory , p. 110