Syntactic monoid

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.

Example

Suppose X₁, ..., 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

• Range: max(X₁, ..., X_n) − min(X₁, ..., X_n)
• Interquartile range: Q₃ − Q₁
• Sample variance:

${\hat {\sigma }}^{2}:=\,{\frac {\sum \left(X_{i}-{\overline {X}}\right)^{2}}{n}}$

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 σ², the sample mean ${\overline {X}}$ is not an ancillary statistic of the variance, as the sampling distribution of the sample mean is N(1, σ²/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. 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 at-bats. 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 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 (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 at-bats (e.g., a batting average of .400, which is very high, based on only five at-bats does not inspire anywhere near as much confidence in the player's ability than a 0.400 average based on 100 at-bats). The number N of at-bats 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

• Basu's theorem
• Prediction interval
• Group family
• Conditionality principle

Notes

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