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Concept
Ancillary statistic
Résumé
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. 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.
This concept was first introduced by Ronald Fisher in the 1920s, but its formal definition was only provided in 1964 by Debabrata Basu.
Suppose X1, ..., Xn are independent and identically distributed, and are normally distributed with unknown expected value μ and known variance 1. Let
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.
In a location family of distributions, is an ancillary statistic.
In a scale family of distributions, is an ancillary statistic.
In a location-scale family of distributions, , where is the sample variance, is an ancillary statistic.
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.
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