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The equivariance (see below) of some decision rule in a statistical problem, the statement of which admits of a group  $  G $
of symmetries, under this group  $  G $. 
The notion of invariance of a statistical procedure arises in the first instance in so-called parametric problems of mathematical statistics, when there is a priori information: the probability distribution  $  P ( d \omega ) $
of the outcomes  $  \omega $
of an observation belongs to a known family  $  \{ {P _  \theta  } : {\theta \in \Theta } \} $. 
A statistical decision problem is said to be  $  G $-
equivariant under a group  $  G $
of measurable transformations  $  g $
of a measurable space  $  ( \Omega , B _  \Omega  ) $
of outcomes if the following conditions hold: 1) there is a homomorphism  $  f $
of  $  G $
onto a group  $  \overline{G}\; $
of transformations of the parameter space  $  \Theta $,

$$ 
f :  g  \rightarrow  \overline{g}\;  \in  \overline{G}\; ,\  \forall g \in G ,
$$

with the property

$$ 
( P _  \theta  g ) ( \cdot )  =  P _ {\overline{g}\; ( \theta ) }  ( \cdot ) ,\ \ 
\forall g \in G ;
$$

2) there exists a homomorphism  $  h $
of  $  G $
onto a group  $  \widehat{G}  $
of measurable transformations of a measurable space  $  ( D , B _ {D} ) $
of decisions  $  d $,

$$ 
h :  g  \rightarrow  \widehat{g}   \in  \widehat{G}  ,\  \forall g \in G ,
$$

with the property

$$ 
L ( \overline{g}\; ( \theta ) , \widehat{g}  ( d ) )  =  L ( \theta , d ) ,
$$

where  $  L ( \theta , d ) $
is the loss function; and 3) all the additional a priori information on the possible values of the parameter (the a priori density  $  p ( \theta ) $, 
the subdivision into alternatives  $  \Theta = \Theta _ {1} \cup \dots \cup \Theta _ {s} $, 
etc.) is  $  G $-
invariant or  $  G $-
equivariant. Under these conditions, the decision rule  $  \delta :  \omega \rightarrow \delta ( \omega ) \in D $, 
whether deterministic or random, is called an invariant (more precisely, a  $  G $-
equivariant) procedure if

$$ 
\delta ( g ( \omega ) )  =  \widehat{g}  ( \delta ( \omega ) ) ,\ \ 
\forall \omega \in \Omega ,\  \forall g \in G .
$$

The risk

$$ 
r _  \delta  ( \theta )  =  {\mathsf E} _  \theta  L ( \theta , \delta (
\omega ) )
$$

of an equivariant decision procedure  $  \delta $
is  $  G $-
invariant; in particular, it does not depend on  $  \theta $
if the group  $  G $
acts transitively on  $  \Theta $.

In parametric problems there is, in general, no guaranteed optimal decision procedure which minimizes the risk for each value of the parameter  $  \theta \in \Theta $. 
In particular, a procedure may lead to very small values of the risk for certain values of  $  \theta $
at the expense of worsening the quality for other equally-possible a priori values of the parameter. Equivariance guarantees to some extent that the approach is unbiased. When the group  $  G $
is sufficiently rich, there is an optimal invariant procedure with a uniformly minimal risk among the invariant procedures.

Invariant procedures are widely applied in hypotheses testing (see also [[Invariant test|Invariant test]]) and in the estimation of the parameters of a probability distribution. Thus, in the problem of estimating an unknown vector of means for the family of  $  m $-
dimensional normal distributions

$$ 
p ( \mathbf x , \pmb\alpha )  =  
\frac{1}{( 2 \pi )  ^ {m/2} }
   \mathop{\rm exp}
\left [ 
\frac{- \sum _ {j} ( x _ {j} - \alpha _ {j} )  ^ {2} }{2}
 \right ]
$$

with unit covariance matrix and quadratic loss function  $  \sum _ {j} ( \delta _ {j} - \alpha _ {j} )  ^ {2} $, 
the optimal equivariant estimator is the ordinary sample mean

$$ 
\mathbf x  ^ {*}  =  
\frac{\mathbf x  ^ {(} 1) + \dots + \mathbf x  ^ {(} N) }{N}
 .
$$

Here the group  $  G $
is given by the product of the group  $  S _ {N} $
of permutations of the observations and the group  $   \mathop{\rm Ort} ( m ) $
of motions of the Euclidean space  $  \mathbf R  ^ {m} $; 
$  \overline{G}\; = \widehat{G}  =  \mathop{\rm Ort} ( m) $. 
For  $  m \geq  3 $, 
there exist for this problem non-equivariant estimators leading to a smaller risk than for  $  \mathbf x  ^ {*} $
for all  $  \pmb\alpha $; 
however, the region of essential  "superefficiency"  turns out to be insignificant and diminishes without bound as the size  $  N $
of the sample increases. The possibility of superefficient procedures is connected with the non-compactness of  $  G $.

Equivariant statistical procedures also arise in a number of non-parametric statistical problems, when the a priori family of distributions  $  P $
of outcomes is essentially infinite-dimensional, as well as in the construction of confidence sets for the parameter  $  \theta $
of the distribution in the presence of nuisance parameters.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E.L. Lehmann,   "Testing statistical hypotheses" , Wiley  (1986)</TD></TR></table>