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A [[Statistical test|statistical test]] of size (level) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095080/u0950801.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095080/u0950802.png" />, for testing a compound hypothesis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095080/u0950803.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095080/u0950804.png" /> against a compound alternative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095080/u0950805.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095080/u0950806.png" />, whose power function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095080/u0950807.png" /> (cf. [[Power function of a test|Power function of a test]]) satisfies
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A [[statistical test]] of size (level) $\alpha$, $0 < \alpha < 1$, for testing a compound hypothesis $H_0$: $\theta\in\Theta_0\subset\Theta$ against the compound alternative $H_1$: $\theta\in\Theta_1=\Theta\setminus\Theta_0$, whose [[Power function of a test|power function]] $\beta({\cdot})$ satisfies
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$$
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\beta(\theta) \le \alpha \ \ \text{if}\ \ \theta \in \Theta_0 \,,
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$$
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$$
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\beta(\theta) \ge \alpha \ \ \text{if}\ \ \theta \in \Theta_1 \ .
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$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095080/u0950808.png" /></td> </tr></table>
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In many problems in statistical hypotheses testing there are no [[uniformly most-powerful test]]s. But if one restricts the class of tests, then there may be uniformly most-powerful tests in that class. If in the problem of testing the hypothesis $H_0$ against the alternative $H_1$ there is a uniformly most-powerful test, then it is unbiased (cf. Unbiased test), since the power of such a test cannot be less than that of the so-called trivial test whose critical function $\Phi({\cdot})$ is constant and equal to the size $\alpha$, that is, $\Phi(X) = \alpha$, where $X$ is the random variable whose realization is used to test the hypothesis $H_0$ against the alternative $H_1$.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095080/u0950809.png" /></td> </tr></table>
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Example. The [[sign test]] is uniformly most-powerful unbiased in the problem of testing the hypothesis $H_0$ according to which the unknown true value of the parameter $p$ of the [[binomial distribution]] is equal to $\frac12$ against the alternative $H_1$: $p\ne\frac12$.
  
In many problems in statistical hypotheses testing there are no uniformly most-powerful tests (cf. [[Uniformly most-powerful test|Uniformly most-powerful test]]). But if one restricts the class of tests, then there may be uniformly most-powerful tests in that class. If in the problem of testing the hypothesis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095080/u09508010.png" /> against the alternative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095080/u09508011.png" /> there is a uniformly most-powerful test, then it is unbiased (cf. Unbiased test), since the power of such a test cannot be less than that of the so-called trivial test whose critical function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095080/u09508012.png" /> is constant and equal to the size <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095080/u09508013.png" />, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095080/u09508014.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095080/u09508015.png" /> is the random variable whose realization is used to test the hypothesis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095080/u09508016.png" /> against the alternative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095080/u09508017.png" />.
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====References====
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<table>
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<TR><TD valign="top">[1]</TD> <TD valign="top">  E.L. Lehmann,   "Testing statistical hypotheses" , Wiley  (1959)</TD></TR>
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</table>
  
Example. The [[Sign test|sign test]] is uniformly most-powerful unbiased in the problem of testing the hypothesis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095080/u09508018.png" /> according to which the unknown true value of the parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095080/u09508019.png" /> of the binomial distribution is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095080/u09508020.png" /> against the alternative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095080/u09508021.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095080/u09508022.png" />.
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{{TEX|done}}
 
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E.L. Lehmann,  "Testing statistical hypotheses" , Wiley  (1959)</TD></TR></table>
 

Latest revision as of 21:08, 27 December 2017

A statistical test of size (level) $\alpha$, $0 < \alpha < 1$, for testing a compound hypothesis $H_0$: $\theta\in\Theta_0\subset\Theta$ against the compound alternative $H_1$: $\theta\in\Theta_1=\Theta\setminus\Theta_0$, whose power function $\beta({\cdot})$ satisfies $$ \beta(\theta) \le \alpha \ \ \text{if}\ \ \theta \in \Theta_0 \,, $$ $$ \beta(\theta) \ge \alpha \ \ \text{if}\ \ \theta \in \Theta_1 \ . $$

In many problems in statistical hypotheses testing there are no uniformly most-powerful tests. But if one restricts the class of tests, then there may be uniformly most-powerful tests in that class. If in the problem of testing the hypothesis $H_0$ against the alternative $H_1$ there is a uniformly most-powerful test, then it is unbiased (cf. Unbiased test), since the power of such a test cannot be less than that of the so-called trivial test whose critical function $\Phi({\cdot})$ is constant and equal to the size $\alpha$, that is, $\Phi(X) = \alpha$, where $X$ is the random variable whose realization is used to test the hypothesis $H_0$ against the alternative $H_1$.

Example. The sign test is uniformly most-powerful unbiased in the problem of testing the hypothesis $H_0$ according to which the unknown true value of the parameter $p$ of the binomial distribution is equal to $\frac12$ against the alternative $H_1$: $p\ne\frac12$.

References

[1] E.L. Lehmann, "Testing statistical hypotheses" , Wiley (1959)
How to Cite This Entry:
Unbiased test. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Unbiased_test&oldid=12993
This article was adapted from an original article by M.S. Nikulin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article