Point estimator

A statistical estimator whose values are points in the set of values of the quantity to be estimated.

Suppose that in the realization $ x = ( x _ {1} \dots x _ {n} ) ^ {T} $ of the random vector $ X = ( X _ {1} \dots X _ {n} ) ^ {T} $, taking values in a sample space $ ( \mathfrak X , {\mathcal B} , {\mathsf P} _ \theta ) $, $ \theta = ( \theta _ {1} \dots \theta _ {k} ) ^ {T} \in \Theta \subset \mathbf R ^ {k} $, the unknown parameter $ \theta $( or some function $ g ( \theta ) $) is to be estimated. Then any statistic $ T _ {n} = T _ {n} ( X) $ producing a mapping of the set $ \mathfrak X $ into $ \Theta $( or into the set of values of $ g ( \theta ) $) is called a point estimator of $ \theta $( or of the function $ g ( \theta ) $ to be estimated). Important characteristics of a point estimator $ T _ {n} $ are its mathematical expectation

$$ {\mathsf E} _ \theta \{ T _ {n} \} = \ \int\limits _ {\mathfrak X } T _ {n} ( x) d {\mathsf P} _ \theta ( x) $$

and the covariance matrix

$$ {\mathsf E} _ \theta \{ ( T _ {n} - {\mathsf E} \{ T _ {n} \} ) ( T _ {n} - {\mathsf E} \{ T _ {n} \} ) ^ {T} \} . $$

The vector $ d ( X) = T _ {n} ( X) - g ( \theta ) $ is called the error vector of the point estimator $ T _ {n} $. If

$$ b ( \theta ) = \ {\mathsf E} _ \theta \{ d ( X) \} = \ {\mathsf E} _ \theta \{ T _ {n} \} - g ( \theta ) $$

is the zero vector for all $ \theta \in \Theta $, then one says that $ T _ {n} $ is an unbiased estimator of $ g ( \theta ) $ or that $ T _ {n} $ is free of systematic errors; otherwise, $ T _ {n} $ is said to be biased, and the vector $ b ( \theta ) $ is called the bias or systematic error of the point estimator. The quality of a point estimator can be defined by means of the risk function (cf. Risk of a statistical procedure).

References

[1]	H. Cramér, "Mathematical methods of statistics" , Princeton Univ. Press (1946)
[2]	I.A. Ibragimov, R.Z. [R.Z. Khas'minskii] Has'minskii, "Statistical estimation: asymptotic theory" , Springer (1981) (Translated from Russian)

References

[a1]	E.L. Lehmann, "Theory of point estimation" , Wiley (1983)