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A theorem stating conditions under which there exists a maximin invariant test in a problem of statistical hypothesis testing.

Suppose that based on the realization of a random variable  $  X $
taking values in a sampling space  $  ( \mathfrak X , {\mathcal B} , {\mathsf P} _  \theta  ) $, 
$  \theta \in \Theta $, 
it is necessary to test a hypothesis  $  H _ {0} $: 
$  \theta \in \Theta _ {0} \subset  \Theta $
against an alternative  $  H _ {1} $: 
$  \theta \in \Theta _ {1} = \Theta \setminus  \Theta _ {0} $, 
and it is assumed that the family  $  \{ {\mathsf P} _  \theta  \} $
is dominated by a certain  $  \sigma $-
finite measure  $  \mu $(
cf. [[Domination|Domination]]). Next, suppose that on  $  ( \mathfrak X , {\mathcal B} ) $
a transformation group  $  G = \{ g \} $
acts that leaves invariant the problem of testing the hypothesis  $  H _ {0} $
against  $  H _ {1} $, 
and let  $  {\mathcal A} $
be the Borel  $  \sigma $-
field of subsets of  $  G $. 
The Hunt–Stein theorem asserts that if the following conditions hold:

1) the mapping  $  ( x, g) \rightarrow gx $
is  $  ( {\mathcal B} \times {\mathcal A} ) $-
measurable and  $  Ag \in {\mathcal B} $
for every set  $  A \in {\mathcal A} $
and any element  $  g \in G $;

2) on  $  A $
there exists an asymptotically right-invariant sequence of measures  $  \nu _ {n} $
in the sense that for any  $  g \in G $
and  $  A \in {\mathcal A} $,

$$ 
\lim\limits _ {n \rightarrow \infty } \ 
| \nu _ {n} ( Ag) - \nu _ {n} ( A) |  =  0;
$$

then for any statistical test intended for testing  $  H _ {0} $
against  $  H _ {1} $
and with critical function  $  \phi ( x) $, 
there is an (almost-) invariant test with critical function  $  \psi ( x) $
such that for all  $  \theta \in \Theta $,

$$ 
\inf _ {\overline{G}\; } \ 
{\mathsf E} _ {\overline{g}\; \theta }  \phi ( X)  \leq  \ 
{\mathsf E} _  \theta  \psi ( X)  \leq  \ 
\sup _ {\overline{G}\; }  {\mathsf E} _ {\overline{g}\; \theta } 
\phi ( X),
$$

where  $  \overline{G}\; = \{ \overline{g}\; \} $
is the group induced by  $  G $.

The Hunt–Stein theorem implies that if there exists a statistical test of level  $  \alpha $
with critical function  $  \phi _ {0} $
that maximizes  $  \inf _ {\theta \in \Theta _ {1}  }  {\mathsf E} _  \theta  \phi _ {0} ( X) $, 
then there also exists an (almost-) invariant test with the same property.

Condition 2) holds necessarily when  $  G $
is a locally compact group on which a right-invariant Haar measure is given. The Hunt–Stein theorem shows that if  $  G $
satisfies the conditions of the theorem, then in any problem of statistical hypothesis testing that is invariant relative to  $  G $
and on which there exists a uniformly most-powerful test, this test is a maximin test.

Conversely, suppose that in some problem of statistical hypotheses testing that is invariant under a group  $  G $
it is established that a uniformly most-powerful test is not a maximin test. This means that the conditions of the Hunt–Stein theorem are violated. In this connection there arises the question: Can a given test be maximin in another problem of hypothesis testing that is invariant under the same group  $  G $? 
The answer to this question depends not only on the group  $  G $, 
but also on the family of distributions  $  \{ {\mathsf P} _  \theta  \} $
itself.

The theorem was obtained by G. Hunt and C. Stein in 1946, see [[#References|[1]]].

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