Nuisance parameter

In statistics, a nuisance parameter is any parameter which is not of immediate interest but which must be accounted for in the analysis of those parameters which are of interest. The classic example of a nuisance parameter is the variance, σ2, of a normal distribution, when the mean, μ, is of primary interest. In statistics, a nuisance parameter is any parameter which is not of immediate interest but which must be accounted for in the analysis of those parameters which are of interest. The classic example of a nuisance parameter is the variance, σ2, of a normal distribution, when the mean, μ, is of primary interest. Nuisance parameters are often variances, but not always; for example in errors-in-variables models, the unknown true location of each observation is a nuisance parameter. In general, any parameter which intrudes on the analysis of another may be considered a nuisance parameter. A parameter may also cease to be a 'nuisance' if it becomes the object of study, as the variance of a distribution may be. The general treatment of nuisance parameters can be broadly similar between frequentist and Bayesian approaches to theoretical statistics. It relies on an attempt to partition the likelihood function into components representing information about the parameters of interest and information about the other (nuisance) parameters. This can involve ideas about sufficient statistics and ancillary statistics. When this partition can be achieved it may be possible to complete a Bayesian analysis for the parameters of interest by determining their joint posterior distribution algebraically. The partition allows frequentist theory to develop general estimation approaches in the presence of nuisance parameters. If the partition cannot be achieved it may still be possible to make use of an approximate partition. In some special cases, it is possible to formulate methods that circumvent the presences of nuisance parameters. The t-test provides a practically useful test because the test statistic does not depend on the unknown variance. It is a case where use can be made of a pivotal quantity. However, in other cases no such circumvention is known.