Ancillary statistic

Remarks

Example: let $X_1, \dots, X_n$ be i.i.d. $\mathcal{N}(\mu, 1)$ with $\mu \in \R$ and $n \ge 2$. Then $X_i - \mu$ are i.i.d. $\mathcal{N}(0, 1)$, and the sample variance $S^2 = \frac{1}{n-1}\sum_i (X_i - \bar{X})^2$ is a function of them alone, because $X_i - \bar{X} = (X_i - \mu) - (\bar{X} - \mu)$; so $S^2$ is ancillary for $\mu$. Non-example: in the same model, $\bar{X} \sim \mathcal{N}(\mu, 1/n)$, whose law changes with $\mu$, so $\bar{X}$ is not ancillary. In a location model $X_i = \mu + \varepsilon_i$ with a fixed law of the $\varepsilon_i$, every function of the differences $X_i - X_j$ is ancillary. In a scale model $X_i = \sigma \varepsilon_i$, every function of the ratios $X_i / X_j$ is ancillary, when these are defined.