Local Ancillarity in the Presence of a Nuisance Parameter

Approximate ancillarity in the presence of a nuisance parameter is considered. A theory of approximate S-ancillarity, based on the concept of local ancillarity introduced by Cox (1980), is developed. First- and second-order local S-ancillarity are considered in detail and these results are applied to several examples. These examples indicate that, although first-order local S-ancillarity is a weak condition, second-order local S-ancillarity is too strong a requirement for general use. Hence, a second aspect of the paper is a study of the properties of an approximately ancillary statistic that are desirable for inference in the presence of a nuisance parameter; this leads to a weaker definition of local ancillarity in the presence of a nuisance parameter that is satisfied in many cases.