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
the risk of the point estimator at the point measured with respect to
Then an estimator is called L-unadulterated if it holds for all :
-
for everyone .
L-unadulterated estimators are therefore closer to the value than to any further value with regard to the loss function L, measured with .
Examples
Gaussian loss
If you choose the Gaussian loss as the loss function
-
,
so (see L p -space ) is L-unadulterated if and only if an unbiased estimator is for .
Laplace loss and median integrity
If one chooses the Laplace loss as the loss function
-
,
so is L-unadulterated if and only if the median is unadulterated, that is, it applies to all
-
and .
literature