📂Mathematical Statistics

Auxiliary Statistics

Definition ¹

Let $S$ be a statistic of sample $\mathbf{X}$. If the distribution of $S \left( \mathbf{X} \right)$ does not depend on the parameter $\theta$, it is called an Ancillary Statistic.

Description

Actually, nobody says ancillary statistic in conversation, they pronounce it as [ancillary statistic].

If a sufficient statistic has all the information about $\theta$, then an ancillary statistic can be thought of as a statistic that has no information about $\theta$ at all.

For example, consider a random sample $X_{1} , \cdots , X_{n}$ from a normal distribution $N \left( \mu , \sigma^{2} \right)$. The sample variance $$ S^{2} = {{ 1 } \over { n -1 }} \sum_{k=1}^{n} X_{k}^{2} $$ is a sufficient statistic for the population variance $\sigma^{2}$, but according to Student’s theorem, $$ {{ n-1 } \over { \sigma^{2} }} S^{2} \sim \chi_{n-1}^{2} $$ This means that the population variance $\mu$ does not appear in the chi-squared distribution that the sample variance follows, and it is an ancillary statistic regarding $\mu$.