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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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S. Zachs,   "The theory of statistical inference" , Wiley  (1971)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E.L. Lehmann,   "Testing statistical hypotheses" , Wiley  (1986)</TD></TR></table>