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Any unknown parameter of a [[Probability distribution|probability distribution]] in a statistical problem connected with the study of other parameters of a given distribution. More precisely, for a realization of a random variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067880/n0678801.png" />, taking values in a sample space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067880/n0678802.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067880/n0678803.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067880/n0678804.png" />, suppose it is necessary to make a statistical inference about the parameters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067880/n0678805.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067880/n0678806.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067880/n0678807.png" /> are nuisance parameters in the problem. For example, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067880/n0678808.png" /> be independent random variables, subject to the normal law <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067880/n0678809.png" />, with unknown parameters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067880/n06788010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067880/n06788011.png" />, and one wishes to test the hypothesis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067880/n06788012.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067880/n06788013.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067880/n06788014.png" /> is some fixed number. The unknown variance <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067880/n06788015.png" /> is a nuisance parameter in the problem of testing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067880/n06788016.png" />. Another important example of a problem with a nuisance parameter is the [[Behrens–Fisher problem|Behrens–Fisher problem]]. Naturally, for the solution of a statistical problem with nuisance parameters it is desirable to be able to make a statistical inference not depending on these parameters. In the theory of statistical hypothesis testing one often achieves this by narrowing the class of tests intended for testing a certain hypothesis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067880/n06788017.png" /> in the presence of a nuisance parameter to a class of similar tests (cf. [[Statistical test|Statistical test]]).
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Any unknown parameter of a [[Probability distribution|probability distribution]] in a statistical problem connected with the study of other parameters of a given distribution. More precisely, for a realization of a random variable $  X $,  
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taking values in a sample space $  ( \mathfrak X , \mathfrak B , {\mathsf P} _  \theta  ) $,  
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$  \theta = ( \theta _ {1} \dots \theta _ {n} ) $,  
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$  \theta \in \mathbf R  ^ {n} $,  
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suppose it is necessary to make a statistical inference about the parameters $  \theta _ {1} \dots \theta _ {k} $,  
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$  k < n $.  
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Then $  \theta _ {k+} 1 \dots \theta _ {n} $
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are nuisance parameters in the problem. For example, let $  X _ {1} \dots X _ {n} $
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be independent random variables, subject to the normal law $  \phi ( ( x - \xi ) / \sigma ) $,  
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with unknown parameters $  \xi $
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and $  \sigma  ^ {2} $,  
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and one wishes to test the hypothesis $  H _ {0} $:  
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$  \xi = \xi _ {0} $,  
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where $  \xi _ {0} $
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is some fixed number. The unknown variance $  \sigma  ^ {2} $
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is a nuisance parameter in the problem of testing $  H _ {0} $.  
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Another important example of a problem with a nuisance parameter is the [[Behrens–Fisher problem|Behrens–Fisher problem]]. Naturally, for the solution of a statistical problem with nuisance parameters it is desirable to be able to make a statistical inference not depending on these parameters. In the theory of statistical hypothesis testing one often achieves this by narrowing the class of tests intended for testing a certain hypothesis $  H _ {0} $
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in the presence of a nuisance parameter to a class of similar tests (cf. [[Statistical test|Statistical test]]).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  Yu.V. Linnik,  "Statistical problems with nuisance parameters" , Amer. Math. Soc.  (1968)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  Yu.V. Linnik,  "Statistical problems with nuisance parameters" , Amer. Math. Soc.  (1968)  (Translated from Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E.L. Lehmann,  "Testing statistical hypotheses" , Wiley  (1978)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  E.L. Lehmann,  "Theory of point estimation" , Wiley  (1983)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E.L. Lehmann,  "Testing statistical hypotheses" , Wiley  (1978)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  E.L. Lehmann,  "Theory of point estimation" , Wiley  (1983)</TD></TR></table>

Latest revision as of 08:03, 6 June 2020


Any unknown parameter of a probability distribution in a statistical problem connected with the study of other parameters of a given distribution. More precisely, for a realization of a random variable $ X $, taking values in a sample space $ ( \mathfrak X , \mathfrak B , {\mathsf P} _ \theta ) $, $ \theta = ( \theta _ {1} \dots \theta _ {n} ) $, $ \theta \in \mathbf R ^ {n} $, suppose it is necessary to make a statistical inference about the parameters $ \theta _ {1} \dots \theta _ {k} $, $ k < n $. Then $ \theta _ {k+} 1 \dots \theta _ {n} $ are nuisance parameters in the problem. For example, let $ X _ {1} \dots X _ {n} $ be independent random variables, subject to the normal law $ \phi ( ( x - \xi ) / \sigma ) $, with unknown parameters $ \xi $ and $ \sigma ^ {2} $, and one wishes to test the hypothesis $ H _ {0} $: $ \xi = \xi _ {0} $, where $ \xi _ {0} $ is some fixed number. The unknown variance $ \sigma ^ {2} $ is a nuisance parameter in the problem of testing $ H _ {0} $. Another important example of a problem with a nuisance parameter is the Behrens–Fisher problem. Naturally, for the solution of a statistical problem with nuisance parameters it is desirable to be able to make a statistical inference not depending on these parameters. In the theory of statistical hypothesis testing one often achieves this by narrowing the class of tests intended for testing a certain hypothesis $ H _ {0} $ in the presence of a nuisance parameter to a class of similar tests (cf. Statistical test).

References

[1] Yu.V. Linnik, "Statistical problems with nuisance parameters" , Amer. Math. Soc. (1968) (Translated from Russian)

Comments

References

[a1] E.L. Lehmann, "Testing statistical hypotheses" , Wiley (1978)
[a2] E.L. Lehmann, "Theory of point estimation" , Wiley (1983)
How to Cite This Entry:
Nuisance parameter. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Nuisance_parameter&oldid=15002
This article was adapted from an original article by M.S. Nikulin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article