Ancillary statistic
An ancillary statistic is a
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, ..., Xn are
be the sample mean.
The following statistical measures of dispersion of the sample
- Range: max(X1, ..., Xn) − min(X1, ..., Xn)
- Interquartile range: Q3 − Q1
- Sample variance:
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 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
In a scale family of distributions, is an ancillary statistic.
In a
In recovery of information
It turns out that, if is a non-sufficient statistic and is ancillary, one can sometimes recover all the information about the unknown parameter contained in the entire data by reporting while conditioning on the observed value of . This is known as conditional inference.[3]
For example, suppose that follow the distribution where is unknown. Note that, even though is not sufficient for (since its Fisher information is 1, whereas the Fisher information of the complete statistic is 2), by additionally reporting the ancillary statistic , one obtains a joint distribution with Fisher information 2.[3]
Ancillary complement
Given a statistic T that is not
The statistic is particularly useful if one takes T to be a
Example
In
- 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
See also
Notes
- JSTOR 4355624.
- ^ JSTOR 24309506.
- ^ ISBN 0-8247-0379-0.
- ISBN 978-1-4419-5825-9
- ISSN 0305-0041.
- JSTOR 25049300.
- ISBN 978-0-940600-50-8, retrieved 2023-04-24
- ISSN 0162-1459.
- ^ "Ancillary statistics" (PDF).