are all ancillary statistics, because their sampling distributions do not change as μ changes. Computationally, this is because in the formulas, the μ terms cancel – adding a constant number to a distribution changes its sample maximum and minimum by the same amount, so it does not change their difference, and likewise for others: these measures of dispersion do not depend on location. Conversely, given i.i.d. normal variables with known mean 1 and unknown variance σ2, the sample mean is not an ancillary statistic of the variance, as the sampling distribution of the sample mean is N, which does depend on σ2 – this measure of location depends on dispersion.
Ancillary complement
Given a statistic T that is not sufficient, an ancillary complement is a statistic U that is ancillary and such that is sufficient. Intuitively, an ancillary complement "adds the missing information". The statistic is particularly useful if one takes T to be a maximum likelihood estimator, which in general will not be sufficient; then one can ask for an ancillary complement. In this case, Fisher argues that one must condition on an ancillary complement to determine information content: one should consider the Fisher informationcontent of T to not be the marginal of T, but the conditional distribution of T, given U: how much information does Tadd? This is not possible in general, as no ancillary complement need exist, and if one exists, it need not be unique, nor does a maximum ancillary complement exist.
Example
In baseball, suppose a scout observes a batter in N at-bats. Suppose that the number N is chosen by some random process that is independent of the batter's ability – say a coin is tossed after each at-bat and the result determines whether the scout will stay to watch the batter's next at-bat. The eventual data are the number N of at-bats and the number X of hits: the data are a sufficient statistic. The observed batting averageX/N fails to convey all of the information available in the data because it fails to report the number N of at-bats. The number N of at-bats is an ancillary statistic because
It is a part of the observable data, and
Its probability distribution does not depend on the batter's ability, since it was chosen by a random process independent of the batter's ability.
This ancillary statistic is an ancillary complement to the observed batting average X/N, i.e., the batting average X/N is not a sufficient statistic, in that it conveys less than all of the relevant information in the data, but conjoined with N, it becomes sufficient.
