L-authenticity

In estimation theory , a branch of mathematical statistics , L-authenticity is a property of a point estimator . It generalizes the faithfulness to expectations and contains the median authenticity as a further special case . The generalization takes place via the use of a general loss function .

definition

A statistical model and a loss function are given . Be it ${\displaystyle (X,{\mathcal {A}},(P_{\vartheta })_{\vartheta \in \Theta })}$ ${\displaystyle L(\cdot ;\cdot )}$

${\displaystyle R_{P_{\vartheta _{0}}}(\vartheta ,S)=\int _{X}L(g(\vartheta );S)\mathrm {d} P_{\vartheta _{0}}}$

the risk of the point estimator at the point measured with respect to ${\displaystyle S}$ ${\displaystyle \vartheta }$ ${\displaystyle P_{\vartheta _{0}}}$

Then an estimator is called L-unadulterated if it holds for all : ${\displaystyle S}$ ${\displaystyle \vartheta _{0}\in \Theta }$

${\displaystyle R_{P_{\vartheta _{0}}}(\vartheta _{0},S)\leq R_{P_{\vartheta _{0}}}(\vartheta ,S)}$   for everyone   . ${\displaystyle \vartheta \in \Theta }$

L-unadulterated estimators are therefore closer to the value than to any further value with regard to the loss function L, measured with . ${\displaystyle P_{\vartheta _{0}}}$ ${\displaystyle g(\vartheta _{0})}$ ${\displaystyle g(\vartheta )}$

Examples

Gaussian loss

If you choose the Gaussian loss as the loss function

${\displaystyle L(\vartheta ;a)=(g(\vartheta )-a)^{2}}$,

so (see L ^p -space ) is L-unadulterated if and only if an unbiased estimator is for . ${\displaystyle S\in L^{2}((P_{\vartheta })_{\vartheta \in \Theta })}$ ${\displaystyle S}$ ${\displaystyle g}$

Laplace loss and median integrity

If one chooses the Laplace loss as the loss function

${\displaystyle L(\vartheta ;a)=|g(\vartheta )-a|}$,

so is L-unadulterated if and only if the median is unadulterated, that is, it applies to all ${\displaystyle S}$ ${\displaystyle S}$ ${\displaystyle \vartheta \in \Theta }$

${\displaystyle P_{\vartheta }(S\geq g(\vartheta ))\geq {\frac {1}{2}}}$   and   . ${\displaystyle P_{\vartheta }(S\leq g(\vartheta ))\geq {\frac {1}{2}}}$

literature

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