# 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 , the components of which are independent, identically distributed random variables taking values in a sampling space , , it is necessary to estimate the unknown true value of the parameter . Next, suppose that on acts a group of one-to-one transformations such that

In turn, the group generates on the parameter space a so-called induced group of transformations , the elements of which are defined by the formula

Let be a group of one-to-one transformations on such that

Under these conditions it is said that a point estimator of is an equivariant estimator, or that it preserves the structure of the problem of statistical estimation of the parameter with respect to the group , if

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 .

#### 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. M.S. Nikulin (originator),

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Equivariant_estimator&oldid=15622