Let \(S\) be a statistic with density depending on a scale parameter \(\theta\) and nuisance parameter \(\lambda\). If \(S\) can be written as \(S=(T,A)\), then \(A\) is ancillary for \(\theta\) in the presence of \(\lambda\) if the conditional distribution of \(T\) given \(A\) depends only on \(\theta\) and \(A\) contains “no information about \(\theta\) in the presence of \(\lambda\)”.
Here, giving a precise meaning to this last phrase, the author introduces a definition of approximate \(S\)-ancillarity based on the concept of local ancillarity. First- and second-order local \(S\)-ancillarity are defined and the results are applied to several examples (normal mean and variance, exponential families, regression models with additive error).
Since second-order local \(S\)-ancillarity is relatively strong, the author introduces local \(I\)-ancillarity, and finds that second-order locally \(I\)-ancillary statistics exist in many cases in which second-order locally \(S\)-ancillary statistics are difficult to construct. The examples discussed earlier are then considered with respect to second-order local \(I\)-ancillarity.