Nuisance parameter

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)

References

[a1]	E.L. Lehmann, "Testing statistical hypotheses" , Wiley (1978)
[a2]	E.L. Lehmann, "Theory of point estimation" , Wiley (1983)