Lehmann–Scheffé theorem
Encyclopedia
In statistics
, the Lehmann–Scheffé theorem is prominent in mathematical statistics, tying together the ideas of completeness, sufficiency, uniqueness, and best unbiased estimation. The theorem states that any estimator
which is unbiased for a given unknown quantity and which is based on only a complete
, sufficient statistic
(and on no other data-derived values) is the unique best unbiased estimator of that quantity. The Lehmann–Scheffé theorem is named after Erich Leo Lehmann
and Henry Scheffé
, given their two early papers.
Formally, if T is a complete sufficient statistic for θ and E(g(T)) = τ(θ) then g(T) is the minimum-variance unbiased estimator
(MVUE) of τ(θ).
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
, the Lehmann–Scheffé theorem is prominent in mathematical statistics, tying together the ideas of completeness, sufficiency, uniqueness, and best unbiased estimation. The theorem states that any estimator
Estimator
In statistics, an estimator is a rule for calculating an estimate of a given quantity based on observed data: thus the rule and its result are distinguished....
which is unbiased for a given unknown quantity and which is based on only a complete
Completeness (statistics)
In statistics, completeness is a property of a statistic in relation to a model for a set of observed data. In essence, it is a condition which ensures that the parameters of the probability distribution representing the model can all be estimated on the basis of the statistic: it ensures that the...
, sufficient statistic
Sufficiency (statistics)
In statistics, a sufficient statistic is a statistic which has the property of sufficiency with respect to a statistical model and its associated unknown parameter, meaning that "no other statistic which can be calculated from the same sample provides any additional information as to the value of...
(and on no other data-derived values) is the unique best unbiased estimator of that quantity. The Lehmann–Scheffé theorem is named after Erich Leo Lehmann
Erich Leo Lehmann
Erich Leo Lehmann was an American statistician, who contributed to statistical and nonparametric hypothesis testing...
and Henry Scheffé
Henry Scheffé
Henry Scheffé was an American statistician. He is known for the Lehmann–Scheffé theorem and Scheffé's method.- External links :...
, given their two early papers.
Formally, if T is a complete sufficient statistic for θ and E(g(T)) = τ(θ) then g(T) is the minimum-variance unbiased estimator
Minimum-variance unbiased estimator
In statistics a uniformly minimum-variance unbiased estimator or minimum-variance unbiased estimator is an unbiased estimator that has lower variance than any other unbiased estimator for all possible values of the parameter.The question of determining the UMVUE, if one exists, for a particular...
(MVUE) of τ(θ).