Pivotal quantity

In statistics, a pivotal quantity or pivot is a function of observations and unobservable parameters such that the function's probability distribution does not depend on the unknown parameters (including nuisance parameters).^[1] A pivot need not be a statistic — the function and its 'value' can depend on the parameters of the model, but its 'distribution' must not. If it is a statistic, then it is known as an 'ancillary statistic'.

More formally,^[2] let ${\displaystyle X=(X_{1},X_{2},\ldots ,X_{n})}$ be a random sample from a distribution that depends on a parameter (or vector of parameters) ${\displaystyle \theta }$. Let ${\displaystyle g(X,\theta )}$ be a random variable whose distribution is the same for all ${\displaystyle \theta }$. Then ${\displaystyle g}$ is called a 'pivotal quantity' (or simply a 'pivot').

Pivotal quantities are commonly used for normalization to allow data from different data sets to be compared. It is relatively easy to construct pivots for location and scale parameters: for the former we form differences so that location cancels, for the latter ratios so that scale cancels.

Pivotal quantities are fundamental to the construction of test statistics, as they allow the statistic to not depend on parameters – for example, Student's t-statistic is for a normal distribution with unknown variance (and mean). They also provide one method of constructing confidence intervals, and the use of pivotal quantities improves performance of the bootstrap. In the form of ancillary statistics, they can be used to construct frequentist prediction intervals (predictive confidence intervals).