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 taking values in a sampling space , , it is necessary to test a hypothesis : against an alternative : , and it is assumed that the family is dominated by a certain -finite measure (cf. Domination). Next, suppose that on a transformation group acts that leaves invariant the problem of testing the hypothesis against , and let be the Borel -field of subsets of . The Hunt–Stein theorem asserts that if the following conditions hold:

1) the mapping is -measurable and for every set and any element ;

2) on there exists an asymptotically right-invariant sequence of measures in the sense that for any and ,

then for any statistical test intended for testing against and with critical function , there is an (almost-) invariant test with critical function such that for all ,

where is the group induced by .

The Hunt–Stein theorem implies that if there exists a statistical test of level with critical function that maximizes , then there also exists an (almost-) invariant test with the same property.

Condition 2) holds necessarily when is a locally compact group on which a right-invariant Haar measure is given. The Hunt–Stein theorem shows that if satisfies the conditions of the theorem, then in any problem of statistical hypothesis testing that is invariant relative to 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 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 ? The answer to this question depends not only on the group , but also on the family of distributions 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)