Super efficient estimator

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With super-efficient estimators is special point estimates from the estimation theory , a branch of mathematical statistics . The peculiarity is that they fall below the Cramér-Rao limit everywhere. A super-efficient estimator therefore has a really lower mean squared estimation error than all regular and unbiased estimators considered in the Cramér-Rao inequality. The best-known representative of super efficient estimators is the James Stein estimator .

Mathematical definition

A d-parametric statistical model with a dominant measure is given . The Fisher information matrix exists and is invertible . These framework assumptions are necessary because super-efficient estimators only make sense if the right-hand side of the Cramér-Rao inequality exists. ${\displaystyle ({\mathcal {X}},{\mathcal {A}},(P_{\vartheta })_{\vartheta \in \Theta })}$ ${\displaystyle \mu }$ ${\displaystyle I(\vartheta )}$

An estimator for the parameter is called super efficient if the following applies to all : ${\displaystyle T:{\mathcal {X}}\to \mathbb {R} ^{d}}$ ${\displaystyle \vartheta \in \Theta \subset \mathbb {R} ^{d}}$ ${\displaystyle \vartheta }$

${\displaystyle \mathrm {E} _{\vartheta }\left[\|T-\vartheta \|^{2}\right]<\operatorname {Spur} (I(\vartheta )^{-1})}$

These are at the track formation of a matrix. ${\displaystyle \operatorname {Spur} }$

existence

The Van Trees inequality can be used to prove that super-efficient estimators do not exist for one- or two-parameter models. The existence of higher parameter models is shown by the James Stein estimator mentioned at the beginning.