Hunt-Stein theorem

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)