Lehmann-Scheffé theorem

The set of Lehmann-Scheffé is a key result of the estimation theory , a branch of mathematical statistics . The statement based on Rao-Blackwell 's theorem provides criteria under which unambiguous point estimators are also consistently best unambiguous estimators, i.e. have a lower variance than all other unambiguous estimators.

The set is named after Erich Leo Lehmann and Henry Scheffé .

statement

Lehmann-Scheffé's theorem can be formulated in different ways, which differ in their notation and the structures used, but are identical in content.

For σ-algebras

A statistical model is given and the set of all unbiased estimators with finite variance for the parameter function . Let the sub- σ-algebra be both sufficient for and complete for . ${\displaystyle (X,{\mathcal {A}},{\mathcal {P}})}$ ${\displaystyle D_{g}}$ ${\displaystyle g}$ ${\displaystyle {\mathcal {S}}\subset {\mathcal {A}}}$ ${\displaystyle {\mathcal {P}}}$ ${\displaystyle {\mathcal {L}}^{2}(X,{\mathcal {A}},{\mathcal {P}})}$

If , then is the Rao-Blackwell improvement of with respect to uniformly best unbiased estimator for . Say it applies ${\displaystyle T\in D_{g}}$ ${\displaystyle T^{+}:=\operatorname {E} _{\bullet }(T|{\mathcal {S}})}$ ${\displaystyle T}$ ${\displaystyle {\mathcal {S}}}$ ${\displaystyle g}$

${\displaystyle \operatorname {Var} _{P}(T^{+})\leq \operatorname {Var} _{P}(K)\quad \mathrm {f{\ddot {u}}r\;alle\;} P\in {\mathcal {P}}}$

and all others . ${\displaystyle K\in D_{g}}$

For statistics

The formulation using statistics follows directly from the above: The sufficient, complete σ-algebra is replaced by sufficient , complete statistics . Part is also noted as. This does not mean that the statement only applies to parametric models. In full formulation, the statement then reads: is a consistently best unbiased estimator for , i.e. it is ${\displaystyle {\mathcal {S}}}$ ${\displaystyle S}$ ${\displaystyle {\mathcal {P}}}$ ${\displaystyle (P_{\vartheta })_{\vartheta \in \Theta }}$ ${\displaystyle T^{+}:=\operatorname {E} _{\bullet }(T|S)}$ ${\displaystyle g}$

${\displaystyle \operatorname {Var} _{\vartheta }(T^{+})\leq \operatorname {Var} _{\vartheta }(K)\quad \mathrm {f{\ddot {u}}r\;alle\;} \vartheta \in \Theta }$

and all others . ${\displaystyle K\in D_{g}}$

Alternative formulations

Possible reformulations of the above statements are:

• If is sufficient and complete for and is , then is equally best unbiased estimator for . ${\displaystyle {\mathcal {S}}}$ ${\displaystyle {\mathcal {L}}^{2}(X,{\mathcal {A}},{\mathcal {P}})}$ ${\displaystyle T\in {\mathcal {L}}^{2}(X,{\mathcal {S}},{\mathcal {P}})\cap D_{g}}$ ${\displaystyle T}$ ${\displaystyle g}$
• If there is a complete sufficient statistic and one exists such that there is an unbiased estimator for , then there is an equally best unbiased estimator for . This is true because . If one now sets in the above statement , this formulation follows. ${\displaystyle T}$ ${\displaystyle h}$ ${\displaystyle h\circ T}$ ${\displaystyle g}$ ${\displaystyle h\circ T}$ ${\displaystyle g}$ ${\displaystyle \operatorname {E} _{\bullet }(h\circ T|T)=h\circ T}$ ${\displaystyle S=h\circ T}$

Generalizations

A specialization of the Lehmann-Scheffé theorem is the Barankin and Stein theorem , which describes the structure of locally minimal estimators .

literature

• Ludger Rüschendorf: Mathematical Statistics . Springer Verlag, Berlin Heidelberg 2014, ISBN 978-3-642-41996-6 , doi : 10.1007 / 978-3-642-41997-3 .
• Claudia Czado, Thorsten Schmidt: Mathematical Statistics . Springer-Verlag, Berlin Heidelberg 2011, ISBN 978-3-642-17260-1 , doi : 10.1007 / 978-3-642-17261-8 .