Barankin and Stein's theorem

The Barankin and Stein theorem is a mathematical theorem of estimation theory , a branch of mathematical statistics . It describes the structure of locally minimal estimators and can thus be regarded as a specialization of the Lehmann-Scheffé theorem , which describes the structure of uniformly best unbiased estimators .

The set is named after Charles Stein and Edward William Barankin .

statement

Framework

A statistical model is given . Be a firm chosen. Furthermore , the distribution class dominates , i.e. each has a density function ${\displaystyle (X,{\mathcal {A}},(P_{\vartheta })_{\vartheta \in \Theta })}$ ${\displaystyle \vartheta _{0}\in \Theta }$ ${\displaystyle P_{\vartheta _{0}}}$ ${\displaystyle (P_{\vartheta })_{\vartheta \in \Theta }}$ ${\displaystyle P_{\vartheta }}$

${\displaystyle f_{\vartheta }:={\frac {\mathrm {d} P_{\vartheta }}{\mathrm {d} P_{\vartheta _{0}}}}}$

regarding . Each of these density functions is assumed to be in relation to the set of all square integrable functions (see Lp space ). ${\displaystyle P_{\vartheta _{0}}}$ ${\displaystyle L^{2}(P_{\vartheta _{0}}):=L^{2}(X,{\mathcal {A}},P_{\vartheta _{0}})}$ ${\displaystyle P_{\vartheta _{0}}}$

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

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

the set of all unbiased estimators with finite variance with respect to . Furthermore, be ${\displaystyle P_{\vartheta _{0}}}$

${\displaystyle \operatorname {span} (A)}$

the linear envelope of the functions in and ${\displaystyle A}$

${\displaystyle {\overline {B}}^{L^{2}(P_{\vartheta _{0}})}}$

concluding the crowd in . ${\displaystyle B}$ ${\displaystyle L^{2}(P_{\vartheta _{0}})}$

sentence

The theorem of Barankin and Stein reads: A is locally optimal in if and only if ${\displaystyle T\in D_{g}(\vartheta _{0})}$ ${\displaystyle \vartheta _{0}}$

${\displaystyle T\in {\overline {\operatorname {span} \left(\{f_{\vartheta }\colon \vartheta \in \Theta \}\right)}}^{L^{2}(P_{\vartheta _{0}})}}$

is.

Evidence sketch

The proof is essentially based on orthogonality arguments in the Hilbert space . With the notation and the scalar product is ${\displaystyle L^{2}(P_{\vartheta _{0}})}$ ${\displaystyle {\mathcal {L}}_{\vartheta }=\operatorname {span} f_{\vartheta }}$ ${\displaystyle \langle \cdot ;\cdot \rangle _{L^{2}(P_{\vartheta _{0}})}}$

${\displaystyle \operatorname {E} _{\vartheta }(T)=0\iff E_{\vartheta _{0}}(Tf_{\vartheta })\iff \langle f_{\vartheta };T\rangle _{L^{2}(P_{\vartheta _{0}})}=0\iff T\in {\mathcal {L}}_{\vartheta }^{\perp }}$.

Hence for , the set of all zero estimators with finite variance with respect to ${\displaystyle D_{0}(\vartheta _{0})}$ ${\displaystyle P_{\vartheta _{0}}}$

${\displaystyle D_{0}(\vartheta _{0})=\bigcap _{\vartheta \in \Theta }{\mathcal {L}}_{\vartheta }^{\perp }}$.

According to the covariance method , however, is locally minimal if and only if is. Since in Hilbert spaces for the orthogonal complement of subspaces ${\displaystyle T}$ ${\displaystyle T\in D_{0}(\vartheta _{0})^{\perp }}$ ${\displaystyle U_{i}}$

${\displaystyle \left(\bigcap _{i\in I}U_{i}^{\perp }\right)^{\perp }={\overline {\operatorname {span} (U_{i})}}}$

holds, follows

${\displaystyle D_{0}(\vartheta _{0})^{\perp }=\left(\bigcap _{\vartheta \in \Theta }{\mathcal {L}}_{\vartheta }^{\perp }\right)^{\perp }={\overline {\operatorname {span} \left(\{f_{\vartheta }\colon \vartheta \in \Theta \}\right)}}^{L^{2}(P_{\vartheta _{0}})}}$.

Using the above statement about the covariance method, the theorem follows.

literature

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