# Hunt-Stein theorem

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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). 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=47280
This article was adapted from an original article by M.S. Nikulin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article