Maximin criterion

maximin test

A statistical test for testing a compound hypothesis $H_0$: $\theta\in\Theta_0\subset\Theta$ against the compound alternative $H_1$: $\theta\in\Theta_1=\Theta\setminus\Theta_0$, whose minimal power (cf. Power of a statistical test) is maximal in the class of all statistical tests intended for testing $H_0$ against $H_1$ and having the same size $\alpha$, $0<\alpha<1$. In statistically testing $H_0$ against $H_1$ a maximin invariant test exists if the problem itself is invariant relative to some group of transformations $G$, and there is a uniformly most-powerful test in the class of corresponding invariant tests (cf. Hunt–Stein theorem). A maximin test exists, in general, if the family of probability distributions determined by the null and competing hypotheses $H_0$ and $H_1$ is absolutely continuous relative to some $\sigma$-finite measure.

References

[1]	E.L. Lehmann, "Testing statistical hypotheses" , Wiley (1959)
[2]	J. Hájek, Z. Sidák, "Theory of rank tests" , Acad. Press (1967)