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Difference between revisions of "Critical level"

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The difference between one and the [[Critical function|critical function]]. Suppose that a certain hypothesis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027080/c0270801.png" /> concerning the distribution of a random variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027080/c0270802.png" /> is being tested, using a test based on a statistic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027080/c0270803.png" /> the distribution function of which — provided <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027080/c0270804.png" /> is true — is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027080/c0270805.png" />. If the critical region for the test is defined by an equality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027080/c0270806.png" />, then the critical level is given by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027080/c0270807.png" />.
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The difference between one and the [[Critical function|critical function]]. Suppose that a certain hypothesis $H_0$ concerning the distribution of a random variable $X$ is being tested, using a test based on a statistic $T(X)$ the distribution function of which — provided $H_0$ is true — is $G(t)$. If the critical region for the test is defined by an equality $T(X)>t$, then the critical level is given by $1-G\{T(X)\}$.
  
 
====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></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></table>

Latest revision as of 15:52, 22 July 2014

The difference between one and the critical function. Suppose that a certain hypothesis $H_0$ concerning the distribution of a random variable $X$ is being tested, using a test based on a statistic $T(X)$ the distribution function of which — provided $H_0$ is true — is $G(t)$. If the critical region for the test is defined by an equality $T(X)>t$, then the critical level is given by $1-G\{T(X)\}$.

References

[1] E.L. Lehmann, "Testing statistical hypotheses" , Wiley (1959)
[2] J. Hájek, Z. Sidák, "Theory of rank tests" , Acad. Press (1967)
How to Cite This Entry:
Critical level. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Critical_level&oldid=32505
This article was adapted from an original article by M.S. Nikulin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article