Ancillary statistic

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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]


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

[math]\displaystyle{ \overline{X}_n = \frac{X_1+\,\cdots\,+X_n}{n} }[/math]

be the sample mean.

The following statistical measures of dispersion of the sample

[math]\displaystyle{ \hat{\sigma}^2:=\,\frac{\sum \left(X_i-\overline{X}\right)^2}{n} }[/math]

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 [math]\displaystyle{ \overline{X} }[/math] 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, [math]\displaystyle{ (X_1 - X_n, X_2 - X_n, \dots, X_{n-1} - X_n) }[/math] is an ancillary statistic.

In a scale family of distributions, [math]\displaystyle{ (\frac{X_1}{X_n}, \frac{X_2}{X_n}, \dots, \frac{X_{n-1}}{X_n}) }[/math] is an ancillary statistic.

In a location-scale family of distributions, [math]\displaystyle{ (\frac{X_1 - X_n}{S}, \frac{X_2 - X_n}{S}, \dots, \frac{X_{n - 1} - X_n}{S}) }[/math], where [math]\displaystyle{ S^2 }[/math] is the sample variance, is an ancillary statistic.[3][9]

In recovery of information

It turns out that, if [math]\displaystyle{ T_1 }[/math] is a non-sufficient statistic and [math]\displaystyle{ T_2 }[/math] is ancillary, one can sometimes recover all the information about the unknown parameter contained in the entire data by reporting [math]\displaystyle{ T_1 }[/math] while conditioning on the observed value of [math]\displaystyle{ T_2 }[/math]. This is known as conditional inference.[3]

For example, suppose that [math]\displaystyle{ X_1, X_2 }[/math] follow the [math]\displaystyle{ N(\theta, 1) }[/math] distribution where [math]\displaystyle{ \theta }[/math] is unknown. Note that, even though [math]\displaystyle{ X_1 }[/math] is not sufficient for [math]\displaystyle{ \theta }[/math] (since its Fisher information is 1, whereas the Fisher information of the complete statistic [math]\displaystyle{ \overline{X} }[/math] is 2), by additionally reporting the ancillary statistic [math]\displaystyle{ X_1 - X_2 }[/math], 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 (TU) 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.


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 (XN) 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


  1. Lehmann, E. L.; Scholz, F. W. (1992). "Ancillarity". Lecture Notes-Monograph Series. Institute of Mathematical Statistics Lecture Notes - Monograph Series 17: 32–51. doi:10.1214/lnms/1215458837. ISBN 0-940600-24-2. ISSN 0749-2170. 
  2. 2.0 2.1 Ghosh, M.; Reid, N.; Fraser, D. A. S. (2010). "Ancillary statistics: A review". Statistica Sinica 20 (4): 1309–1332. ISSN 1017-0405. 
  3. 3.0 3.1 3.2 3.3 Mukhopadhyay, Nitis (2000). Probability and Statistical Inference. United States of America: Marcel Dekker, Inc.. pp. 309–318. ISBN 0-8247-0379-0. 
  4. Dawid, Philip (2011), DasGupta, Anirban, ed., "Basu on Ancillarity" (in en), Selected Works of Debabrata Basu (New York, NY: Springer): pp. 5–8, doi:10.1007/978-1-4419-5825-9_2, ISBN 978-1-4419-5825-9 
  5. Fisher, R. A. (1925). "Theory of Statistical Estimation" (in en). Mathematical Proceedings of the Cambridge Philosophical Society 22 (5): 700–725. doi:10.1017/S0305004100009580. ISSN 0305-0041. Bibcode1925PCPS...22..700F. 
  6. Basu, D. (1964). "Recovery of Ancillary Information". Sankhyā: The Indian Journal of Statistics, Series A (1961-2002) 26 (1): 3–16. ISSN 0581-572X. 
  7. Stigler, Stephen M. (2001) (in en), Ancillary history, Institute of Mathematical Statistics Lecture Notes - Monograph Series, Beachwood, OH: Institute of Mathematical Statistics, pp. 555–567, doi:10.1214/lnms/1215090089, ISBN 978-0-940600-50-8,, retrieved 2023-04-24 
  8. Buehler, Robert J. (1982). "Some Ancillary Statistics and Their Properties". Journal of the American Statistical Association 77 (379): 581–589. doi:10.1080/01621459.1982.10477850. ISSN 0162-1459. 
  9. "Ancillary statistics".