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)