Uniformly most-powerful test

A statistical test of given significance level for testing a compound hypothesis $H_0$ against a compound alternative $H_1$, whose power is not less than the power of any other statistical test for testing $H_0$ against $H_1$ of the same significance level (cf. Power of a statistical test).

Suppose that a compound hypothesis $H_0$: $\theta\in\Theta_0\subset\Theta$ has to be tested against the compound alternative $H_1$: $\theta\in\Theta_1=\Theta\setminus\Theta_0$, and there is given an upper bound $\alpha$, $0<\alpha<1$, for the probability of an error of the first kind, made by rejecting $H_0$ when it is in fact true (the number $\alpha$ is called the significance level of the test, and it is said that the test has level $\alpha$). In this way, the restriction on the probability of an error of the first kind reduces the set of tests for testing $H_0$ against $H_1$ to the class of tests of level $\alpha$. In terms of the power function (cf. Power function of a test) $\beta(\theta)$, $\theta\in\Theta=\Theta_0\cup\Theta_1$, a statistical test of fixed significance level $\alpha$ means that

$$\sup_{\theta\in\Theta_0}\beta(\theta)=\alpha.$$

If, in the class of all tests of level $\alpha$ for testing $H_0$ against $H_1$, there is one whose power function $\beta^*(\theta)$ satisfies

$$\sup_{\theta\in\Theta_0}\beta^*(\theta)=\alpha,\quad\beta^*(\theta)\geq\beta(\theta),\quad\theta\in\Theta_1,$$

where $\beta(\theta)$ is the power function of any other test from this class, then this test is called a uniformly most-powerful test of level $\alpha$ for testing $H_0$ against $H_1$. A uniformly most-powerful test is optimal if the comparison is made in terms of the power of tests.

References

[1]	E.L. Lehmann, "Testing statistical hypotheses" , Wiley (1959)