Parameter function (statistics)

A parameter function is in the mathematical statistics a function in a statistical parametric model , each parameter of the family of probability distributions assigning a functional value, which then by means of a point estimate is to be estimated or a range estimator to be covered. The parameter function usually only takes on the technical task of assigning elements of the decision space to the elements of the parameter set . However , the choice of the parametric function is definitely relevant in the context of the fairness of expectation of an estimator, since not every parametric function can be estimated as expected.

definition

A parametric statistical model with a set of parameters and a decision space are given . Then a function is called a parameter function . ${\displaystyle (X,{\mathcal {A}},(P_{\vartheta })_{\vartheta \in \Theta })}$ ${\displaystyle \Theta }$ ${\displaystyle (\Omega ,{\mathcal {A}})}$ ${\displaystyle g\colon \Theta \to \Omega }$

example

The statistical model is given

${\displaystyle (\{0,1\}^{100},{\mathcal {P}}(\{0,1\})^{\otimes 100},\operatorname {Ber} _{\vartheta }^{\otimes 100})}$,

that models the 100-fold coin toss and has the parameter set . By default, one then usually selects a parameter function ${\displaystyle \Theta =[0,1]}$

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

thus provided with as a decision space . Now one can try using point estimators like the sample mean to estimate the value of the parametric function, which in this case is identical to the value of the parameter. An alternative parameter function would be, for example ${\displaystyle \Omega =[0,1]}$ ${\displaystyle {\mathcal {B}}([0,1])}$

${\displaystyle g(\vartheta )=4\vartheta ^{2}}$

in the decision-making space provided with . Now one can try again to estimate the value of the parametric function. Point estimates such as the sample mean, which turned out to be good in the above case, don't have to be good in this case. An estimator always depends on the parameter function used. ${\displaystyle \Omega =[0,4]}$ ${\displaystyle {\mathcal {B}}([0,4])}$

literature

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