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Difference between revisions of "Power of a statistical test"

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The probability with which a [[Statistical test|statistical test]] for testing a simple hypothesis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074220/p0742201.png" /> against a simple hypothesis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074220/p0742202.png" /> rejects <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074220/p0742203.png" /> when in fact <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074220/p0742204.png" /> is true. In the case when the hypothesis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074220/p0742205.png" />, competing with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074220/p0742206.png" /> in the test, is compound (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074220/p0742207.png" /> itself may be either simple or compound, which is written symbolically: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074220/p0742208.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074220/p0742209.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074220/p07422010.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074220/p07422011.png" />), the power of the statistical test for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074220/p07422012.png" /> against <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074220/p07422013.png" /> is defined as the restriction of the power function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074220/p07422014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074220/p07422015.png" />, of this test to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074220/p07422016.png" />.
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The probability with which a [[Statistical test|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074220/p07422017.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074220/p07422018.png" /> against a compound alternative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074220/p07422019.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074220/p07422020.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074220/p07422021.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074220/p07422022.png" /> is the power function of the test (see [[Power function of a test|Power function of a test]]).
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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|Power function of a test]]).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E.L. Lehmann,  "Testing statistical hypotheses" , Wiley  (1959)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J. Hájek,  Z. Sidák,  "Theory of rank tests" , Acad. Press  (1967)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  B.L. van der Waerden,  "Mathematische Statistik" , Springer  (1957)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  H. Cramér,  "Mathematical methods of statistics" , Princeton Univ. Press  (1946)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E.L. Lehmann,  "Testing statistical hypotheses" , Wiley  (1959)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J. Hájek,  Z. Sidák,  "Theory of rank tests" , Acad. Press  (1967)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  B.L. van der Waerden,  "Mathematische Statistik" , Springer  (1957)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  H. Cramér,  "Mathematical methods of statistics" , Princeton Univ. Press  (1946)</TD></TR></table>

Latest revision as of 11:23, 13 August 2014

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=18477
This article was adapted from an original article by M.S. Nikulin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article