Ronald Fisher
In statistics, an ancillary statistic is a statistic whose sampling distribution does not depend on the parameters of the model. An ancillary statistic is a pivotal quantity that is also a statistic. Ancillary statistics can be used to construct prediction intervals.
This concept was introduced by the statistical geneticist Sir Ronald Fisher.
Contents

Example 1

Ancillary complement 2

See also 3

Notes 4
Example
Suppose X_{1}, ..., X_{n} are independent and identically distributed, and are normally distributed with unknown expected value μ and known variance 1. Let

\overline{X}_n = \frac{X_1+\,\cdots\,+X_n}{n}
be the sample mean.
The following statistical measures of dispersion of the sample


\hat{\sigma}^2:=\,\frac{\sum \left(X_i\overline{X}\right)^2}{n}
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 (and all samples) 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 σ^{2}, the sample mean \overline{X} is not an ancillary statistic of the variance, as the sampling distribution of the sample mean is N(1, σ^{2}/n), which does depend on σ ^{2} – this measure of location (specifically, its standard error) 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 (T, U) is sufficient.^{[1]} Intuitively, an ancillary complement "adds the missing information" (without duplicating any).
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 information content of T to not be the marginal of T, but the conditional distribution of T, given U: how much information does T add? 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 atbats. Suppose (unrealistically) 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 atbat and the result determines whether the scout will stay to watch the batter's next atbat. The eventual data are the number N of atbats and the number X of hits: the data (X, N) are a sufficient statistic. The observed batting average X/N fails to convey all of the information available in the data because it fails to report the number N of atbats (e.g., a batting average of 0.40, which is very high, based on only five atbats does not inspire anywhere near as much confidence in the player's ability than a 0.40 average based on 100 atbats). The number N of atbats is an ancillary statistic because

It is a part of the observable data (it is a statistic), 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.
See also
Notes

^ Ancillary Statistics: A Review by M. Ghosh, N. Reid and D.A.S. Fraser
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