Namespaces
Variants
Actions

Power of a statistical test

From Encyclopedia of Mathematics
Revision as of 11:23, 13 August 2014 by Ivan (talk | contribs) (TeX)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search
The printable version is no longer supported and may have rendering errors. Please update your browser bookmarks and please use the default browser print function instead.

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)
How to Cite This Entry:
Power of a statistical test. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Power_of_a_statistical_test&oldid=32889
This article was adapted from an original article by M.S. Nikulin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article