Pivotal quantity

In statistics, a pivotal quantity or pivot is a function of observations and unobservable parameters whose probability distribution does not depend on the unknown parameters (also referred to as nuisance parameters). Note that 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 ${\displaystyle X=(X_{1},X_{2},\ldots ,X_{n})}$ 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).

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