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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 $, 
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|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|Statistical test]]).

====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>

====Comments====

====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>