Locally minimal estimator

A locally minimal estimator , also called a locally optimal estimator , is a special unbiased point estimator in estimation theory , a branch of mathematical statistics . Locally minimal estimators scatter less than all other estimators for a given probability measure , i.e. their variance is minimal. Thus locally minimal estimators are a weakening of uniformly best fair-expectation estimators , which vary less than all other estimators with respect to an entire class of probability measures.

definition

A statistical model and a parameter function to be estimated are given ${\displaystyle (X,{\mathcal {A}},(P_{\vartheta })_{\vartheta \in \Theta })}$

${\displaystyle g:\Theta \to \mathbb {R} }$.

Let be the set of unbiased estimators for and ${\displaystyle D_{g}}$ ${\displaystyle g}$

${\displaystyle D_{g}(\vartheta _{0}):=D_{g}\cap {\mathcal {L}}^{2}(P_{\vartheta _{0}})}$

is the set of all unbiased estimators for finite variance with respect to , where is. ${\displaystyle g}$ ${\displaystyle P_{\vartheta _{0}}}$ ${\displaystyle \vartheta _{0}\in \Theta }$

Then an estimator is called locally minimal in or locally optimal in if that holds for all others ${\displaystyle S\in D_{g}(\vartheta _{0})}$ ${\displaystyle \vartheta _{0}}$ ${\displaystyle \vartheta _{0}}$ ${\displaystyle T\in D_{g}(\vartheta _{0})}$

${\displaystyle \operatorname {E} _{\vartheta _{0}}(S-g(\vartheta _{0}))^{2}\leq \operatorname {E} _{\vartheta _{0}}(T-g(\vartheta _{0}))^{2}}$

is.

Covariance method

The covariance method provides a possibility to construct locally minimal estimators by means of the covariance or to check for a given estimator whether it is locally minimal. Let it denote the set of all zero estimators and the set of all zero estimators with finite variance with respect to . ${\displaystyle D_{0}}$ ${\displaystyle D_{0}(\vartheta _{0}):=D_{0}\cap {\mathcal {L}}^{2}(P_{\vartheta _{0}})}$ ${\displaystyle P_{\vartheta _{0}}}$

If a is then given, then is locally minimal in if and only if it holds for all that ${\displaystyle S\in D_{g}(\vartheta _{0})}$ ${\displaystyle S}$ ${\displaystyle \vartheta _{0}}$ ${\displaystyle N\in D_{0}(\vartheta _{0})}$

${\displaystyle \operatorname {Cov} _{\vartheta _{0}}(S;N)=0}$

is.

More generally, the covariance method can be applied to every linear subspace of the estimator functions. If there is such a linear subspace, then the statement applies to a ${\displaystyle {\mathcal {L}}}$ ${\displaystyle S\in {\mathcal {L}}\cap D_{g}(\vartheta _{0})}$

${\displaystyle \operatorname {Cov} _{\vartheta _{0}}(S;N)=0\quad \mathrm {f{\ddot {u}}r\;alle\;} N\in {\mathcal {L}}\cap D_{0}(\vartheta _{0})}$,

if and only if local is minimal in for . ${\displaystyle S}$ ${\displaystyle \vartheta _{0}}$ ${\displaystyle {\mathcal {L}}\cap D_{g}(\vartheta _{0})}$

Existence and uniqueness

Existence statements for locally minimal estimators are mostly based on functional analytical concepts. The locally minimal estimators correspond exactly to the minima of the functional that by

${\displaystyle T\mapsto \operatorname {E} _{\vartheta _{0}}(T^{2})}$

is defined. The fundamental theorem of the calculus of variations provides a statement of existence, for example . Something concrete can be concluded: Will of dominating , all density functions from (see lp space ) and so there is an estimator , the locally minimal in is. ${\displaystyle (P_{\vartheta })_{\vartheta \in \Theta }}$ ${\displaystyle P_{\vartheta _{0}}}$ ${\displaystyle f_{\vartheta }:={\tfrac {\mathrm {d} P_{\vartheta }}{\mathrm {d} P_{\vartheta _{0}}}}}$ ${\displaystyle L^{2}(P_{\vartheta _{0}})}$ ${\displaystyle D_{g}(\vartheta _{0})\neq \emptyset }$ ${\displaystyle S}$ ${\displaystyle \vartheta _{0}}$

The covariance method provides the uniqueness of a locally minimal estimator: If a locally minimal estimator exists in , then it is - almost certainly uniquely determined. ${\displaystyle \vartheta _{0}}$ ${\displaystyle P_{\vartheta _{0}}}$

Important statements

In addition to the statements for uniformly best unbiased estimators, which also apply pointwise, i.e. for locally minimal estimators, the following statements are important:

• Barankin and Stein's theorem : It characterizes the locally minimal estimator via the completion of the linear combinations of the density functions of the probability measures involved.
• Chapman-Robbins inequality : It allows the variance of an estimator to be estimated and provides a point-wise version of the Cramér-Rao inequality at the limit crossing . ${\displaystyle P_{\vartheta _{0}}}$

literature

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