In statistics, ancillarity is a property of a statistic computed on a sample dataset in relation to a parametric model of the dataset. An ancillary statistic has the same distribution regardless of the value of the parameters and thus provides no information about them.[1] [2] [3] It is opposed to the concept of a complete statistic which contains no ancillary information. It is closely related to the concept of a sufficient statistic which contains all of the information that the dataset provides about the parameters.
A ancillary statistic is a specific case of a pivotal quantity that is computed only from the data and not from the parameters. They can be used to construct prediction intervals. They are also used in connection with Basu's theorem to prove independence between statistics.
This concept was first introduced by Ronald Fisher in the 1920s,[4] but its formal definition was only provided in 1964 by Debabrata Basu.[5]
Suppose X1, ..., Xn are independent and identically distributed, and are normally distributed with unknown expected value μ and known variance 1. Let
\overline{X}n=
| X1+ … +Xn | |
| n |
be the sample mean.
The following statistical measures of dispersion of the sample
max(X1, ..., Xn) - min(X1, ..., Xn)
Q3 - Q1
| ||||
| \hat{\sigma} |
2}{n}
Conversely, given i.i.d. normal variables with known mean 1 and unknown variance σ2, the sample mean
\overline{X}
In a location family of distributions,
(X1-Xn,X2-Xn,...,Xn-1-Xn)
In a scale family of distributions,
\left(
| X1 | |
| Xn |
,
| X2 | |
| Xn |
,...,
| Xn-1 | |
| Xn |
\right)
In a location-scale family of distributions,
(
| X1-Xn | |
| S |
,
| X2-Xn | |
| S |
,...,
| Xn-Xn | |
| S |
)
S2
It turns out that, if
T1
T2
T1
T2
For example, suppose that
X1,X2
N(\theta,1)
\theta
X1
\theta
\overline{X}
X1-X2
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.
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
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.