Ancillary statistic

An ancillary statistic is a measure of a sample whose distribution (or whose pmf or pdf) does not depend on the parameters of the model.^[1][2][3] An ancillary statistic is a pivotal quantity that is also a statistic. Ancillary statistics can be used to construct prediction intervals. They are also used in connection with Basu's theorem to prove independence between statistics.^[4]

This concept was first introduced by Ronald Fisher in the 1920s,^[5] but its formal definition was only provided in 1964 by Debabrata Basu.^[6][7]

Examples

Suppose X₁, ..., X_n are independent and identically distributed, and are normally distributed with unknown expected value μ and known variance 1. Let

${\displaystyle {\overline {X}}_{n}={\frac {X_{1}+\,\cdots \,+X_{n}}{n}}}$

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:

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

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 ${\displaystyle {\overline {X}}}$ 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.^[8]

In location-scale families

In a location family of distributions, ${\displaystyle (X_{1}-X_{n},X_{2}-X_{n},\dots ,X_{n-1}-X_{n})}$ is an ancillary statistic.

In a scale family of distributions, ${\displaystyle ({\frac {X_{1}}{X_{n}}},{\frac {X_{2}}{X_{n}}},\dots ,{\frac {X_{n-1}}{X_{n}}})}$ is an ancillary statistic.

In a location-scale family of distributions, ${\displaystyle ({\frac {X_{1}-X_{n}}{S}},{\frac {X_{2}-X_{n}}{S}},\dots ,{\frac {X_{n-1}-X_{n}}{S}})}$, where ${\displaystyle S^{2}}$ is the sample variance, is an ancillary statistic.^[3][9]

In recovery of information

It turns out that, if ${\displaystyle T_{1}}$ is a non-sufficient statistic and ${\displaystyle T_{2}}$ is ancillary, one can sometimes recover all the information about the unknown parameter contained in the entire data by reporting ${\displaystyle T_{1}}$ while conditioning on the observed value of ${\displaystyle T_{2}}$. This is known as conditional inference.^[3]

For example, suppose that ${\displaystyle X_{1},X_{2}}$ follow the ${\displaystyle N(\theta ,1)}$ distribution where ${\displaystyle \theta }$ is unknown. Note that, even though ${\displaystyle X_{1}}$ is not sufficient for ${\displaystyle \theta }$ (since its Fisher information is 1, whereas the Fisher information of the complete statistic ${\displaystyle {\overline {X}}}$ is 2), by additionally reporting the ancillary statistic ${\displaystyle X_{1}-X_{2}}$, one obtains a joint distribution with Fisher information 2.^[3]

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.^[2] 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 0.400, 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