Ancillary statistic

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]

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]

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.

• Basu's theorem
• Prediction interval
• Group family
• Conditionality principle