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 .
If , then is the Rao-Blackwell improvement of with respect to uniformly best unbiased estimator for . Say it applies
and all others .
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
and all others .
Alternative formulations
Possible reformulations of the above statements are:
- If is sufficient and complete for and is , then is equally best unbiased estimator for .
- 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.
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 .