# Equivariant estimator

A statistical point estimator that preserves the structure of the problem of statistical estimation relative to a given group of one-to-one transformations of a sampling space.

Suppose that in the realization of a random vector $X = ( X _ {1} \dots X _ {n} )$, the components $X _ {1} \dots X _ {n}$ of which are independent, identically distributed random variables taking values in a sampling space $( \mathfrak X , {\mathcal B} , {\mathsf P} _ \theta )$, $\theta \in \Theta \subseteq \mathbf R ^ {k}$, it is necessary to estimate the unknown true value of the parameter $\theta$. Next, suppose that on $\mathfrak X$ acts a group of one-to-one transformations $G = \{ g \}$ such that

$$g \mathfrak X = \mathfrak X \ \textrm{ and } \ g {\mathcal B} _ {\mathfrak X } = {\mathcal B} _ {\mathfrak X } \ \ \textrm{ for all } g \in G .$$

In turn, the group $G$ generates on the parameter space $\Theta$ a so-called induced group of transformations $\overline{G}\; = \{ \overline{g}\; \}$, the elements of which are defined by the formula

$${\mathsf P} _ \theta ( B) = {\mathsf P} _ {\overline{g}\; \theta } ( g B ) \ \textrm{ for all } g \in G ,\ B \in {\mathcal B} _ {\mathfrak X } .$$

Let $\overline{G}\;$ be a group of one-to-one transformations on $\Theta$ such that

$$\overline{g}\; \Theta = \Theta \ \textrm{ for all } \ \overline{g}\; \in \overline{G}\; .$$

Under these conditions it is said that a point estimator $\widehat \theta = \widehat \theta ( X)$ of $\theta$ is an equivariant estimator, or that it preserves the structure of the problem of statistical estimation of the parameter $\theta$ with respect to the group $G$, if

$$\widehat \theta ( g X ) = \overline{g}\; \widehat \theta ( X) \ \ \textrm{ for all } g \in G .$$

The most interesting results in the theory of equivariant estimators have been obtained under the assumption that the loss function is invariant with respect to $G$.

#### References

 [1] S. Zachs, "The theory of statistical inference" , Wiley (1971) [2] E.L. Lehmann, "Testing statistical hypotheses" , Wiley (1986)
How to Cite This Entry:
Equivariant estimator. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Equivariant_estimator&oldid=46846
This article was adapted from an original article by M.S. Nikulin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article