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A theorem stating conditions under which there exists a maximin invariant test in a problem of statistical hypothesis testing.
 
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048130/h0481301.png" /> taking values in a sampling space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048130/h0481302.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048130/h0481303.png" />, it is necessary to test a hypothesis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048130/h0481304.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048130/h0481305.png" /> against an alternative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048130/h0481306.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048130/h0481307.png" />, and it is assumed that the family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048130/h0481308.png" /> is dominated by a certain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048130/h0481309.png" />-finite measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048130/h04813010.png" /> (cf. [[Domination|Domination]]). Next, suppose that on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048130/h04813011.png" /> a transformation group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048130/h04813012.png" /> acts that leaves invariant the problem of testing the hypothesis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048130/h04813013.png" /> against <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048130/h04813014.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048130/h04813015.png" /> be the Borel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048130/h04813016.png" />-field of subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048130/h04813017.png" />. The Hunt–Stein theorem asserts that if the following conditions hold:
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048130/h04813018.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048130/h04813019.png" />-measurable and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048130/h04813020.png" /> for every set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048130/h04813021.png" /> and any element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048130/h04813022.png" />;
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048130/h04813023.png" /> there exists an asymptotically right-invariant sequence of measures <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048130/h04813024.png" /> in the sense that for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048130/h04813025.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048130/h04813026.png" />,
+
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} $,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048130/h04813027.png" /></td> </tr></table>
+
$$
 +
\lim\limits _ {n \rightarrow \infty } \
 +
| \nu _ {n} ( Ag) - \nu _ {n} ( A) |  = 0;
 +
$$
  
then for any statistical test intended for testing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048130/h04813028.png" /> against <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048130/h04813029.png" /> and with critical function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048130/h04813030.png" />, there is an (almost-) invariant test with critical function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048130/h04813031.png" /> such that for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048130/h04813032.png" />,
+
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 $,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048130/h04813033.png" /></td> </tr></table>
+
$$
 +
\inf _ {\overline{G}\; } \
 +
{\mathsf E} _ {\overline{g}\; \theta }  \phi ( X)  \leq  \
 +
{\mathsf E} _  \theta  \psi ( X)  \leq  \
 +
\sup _ {\overline{G}\; }  {\mathsf E} _ {\overline{g}\; \theta }
 +
\phi ( X),
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048130/h04813034.png" /> is the group induced by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048130/h04813035.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048130/h04813036.png" /> with critical function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048130/h04813037.png" /> that maximizes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048130/h04813038.png" />, then there also exists an (almost-) invariant test with the same property.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048130/h04813039.png" /> is a locally compact group on which a right-invariant Haar measure is given. The Hunt–Stein theorem shows that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048130/h04813040.png" /> satisfies the conditions of the theorem, then in any problem of statistical hypothesis testing that is invariant relative to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048130/h04813041.png" /> and on which there exists a uniformly most-powerful test, this test is a maximin test.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048130/h04813042.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048130/h04813043.png" />? The answer to this question depends not only on the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048130/h04813044.png" />, but also on the family of distributions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048130/h04813045.png" /> itself.
+
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]]].
 
The theorem was obtained by G. Hunt and C. Stein in 1946, see [[#References|[1]]].

Latest revision as of 22:11, 5 June 2020


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). 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 [1].

References

[1] E.L. Lehmann, "Testing statistical hypotheses" , Wiley (1986)
[2] S. Zachs, "The theory of statistical inference" , Wiley (1971)
How to Cite This Entry:
Hunt-Stein theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hunt-Stein_theorem&oldid=22595
This article was adapted from an original article by M.S. Nikulin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article