Let \(X_{1}, ..., X_{n}\) be a random sample from a distribution that depends on a parameter (or vector of parameters) \(\theta\). Let \(g(X,\theta)\) be a random variable whose distribution is the same for all \(\theta\). Then \(g(X,\theta)\) is called a pivotal quantity (or simply a pivot).
A pivotal quantity (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 pivotal quantity need not be a statistic (the function and its value can depend on the parameters of the model, but its distribution must not) but if it is a statistic, then it is known as an [[ancillary-statistic|ancillary statistic]].
Back to top