Power of a statistical test

The probability with which a statistical test for testing a simple hypothesis $H_0$ against a simple hypothesis $H_1$ rejects $H_0$ when in fact $H_1$ is true. In the case when the hypothesis $H_1$, competing with $H_0$ in the test, is compound ($H_0$ itself may be either simple or compound, which is written symbolically: $H_0$: $\theta\in\Theta_0\subset\Theta$, $H_1$: $\theta\in\Theta_1=\Theta\setminus\Theta_0$), the power of the statistical test for $H_0$ against $H_1$ is defined as the restriction of the power function $\beta(\theta)$, $\theta\in\Theta=\Theta_0\cup\Theta_1$, of this test to $\Theta_1$.

In addition, this definition has been broadly generalized to the following: The power of a statistical test for testing $H_0$: $\theta\in\Theta_0\subset\Theta$ against a compound alternative $H_1$: $\theta\in\Theta_1=\Theta\setminus\Theta_0$ is $\inf_{\theta\in\Theta_1}\beta(\theta)$, where $\beta(\theta)$ is the power function of the test (see Power function of a test).

References

[1]	E.L. Lehmann, "Testing statistical hypotheses" , Wiley (1959)
[2]	J. Hájek, Z. Sidák, "Theory of rank tests" , Acad. Press (1967)
[3]	B.L. van der Waerden, "Mathematische Statistik" , Springer (1957)
[4]	H. Cramér, "Mathematical methods of statistics" , Princeton Univ. Press (1946)