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)