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). A pivot quantity 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, let be a random sample from a distribution that depends on a parameter (or vector of parameters) . Let be a random variable whose distribution is the same for all . Then 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). Prediction interval#Normal distribution One of the simplest pivotal quantities is the z-score; given a normal distribution with mean and variance , and an observation x, the z-score: has distribution – a normal distribution with mean 0 and variance 1. Similarly, since the n-sample sample mean has sampling distribution the z-score of the mean also has distribution Note that while these functions depend on the parameters – and thus one can only compute them if the parameters are known (they are not statistics) – the distribution is independent of the parameters.