View source for Nuisance parameter

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]]).

====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>