in statistics, an ancillary statistic is a statistic whose sampling distribution 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. this concept was introduced by the statistical geneticist sir ronald fisher. ==example== suppose x1, ..., x'n are independent and identically distributed, and are normally distributed with unknown expected value Î¼ and known variance 1. let :\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) &minus; min(x1, ..., xn) *interquartile range: q3 &minus; q1 *sample variance: :: \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 Î¼ changes. computationally, this is because in the formulas, the Î¼ terms cancel â 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 Ï2, the sample mean \overline{x} is not an ancillary statistic of the variance, as the sampling distribution of the sample mean is n(1,&nbsp;&sigma;2/n), which does depend on Ï 2 â this measure of location (specifically, its standard error) depends on dispersion. ==ancillary complement== given a statistic t that is not sufficient, an ancillary complement is a statistic u that is ancillary and such that (t,&nbsp;u) is sufficient.ancillary statistics: a review by m. ghosh, n. reid and d.a.s. fraser 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. ===example=== in baseball, suppose a scout obse
Statistical_theory