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)