📂Mathematical Statistics

Definition of a Pivot in Mathematical Statistics

Definition ¹

A random variable $Q \left( \mathbf{X} ; \theta \right) := Q \left( X_{1} , \cdots , X_{n} ; \theta \right)$ whose probability distribution is independent of all parameters $\theta$ is called a pivot or pivotal quantity.

Description

Naturally, $Q$ is a statistic.

The statement that the probability distribution is independent of all parameters $\theta$ means that the cumulative distribution function $F \left( \mathbf{x} ; \theta \right)$ of $Q \left( \mathbf{X} ; \theta \right)$ is the same for all $\theta$. At first glance, the definition might make one curious why $\theta$, which is supposed to be independent anyway, is included in the function, but the following explanation should clear it up at once.

Location-Scale Family

Given the probability density function $f(x;\mu,\sigma)$ of a location-scale family, its pivot is as follows: $$ Q \left( X_{1} , \cdots , X_{n} ; \mu, \sigma \right) = {{ \overline{X} - \mu } \over { \sigma }} $$ This function $Q$ explicitly shows $\mu$ and $\sigma$, but as a result, it is counterbalanced/cancelled out, making the distribution of $Q$ itself independent of $\mu$, $\sigma$.

Pivoting

$$ \left\{ \mu_{0} : -1.96 \le {{ \mathbf{x} - \mu_{0} } \over { \sigma }} \le 1.96 \right\} $$ For example, in a situation where a normal distribution can be assumed, the confidence interval can be expressed as above. In this context, presenting the $C \left( \mathbf{x} \right)$ confidence interval itself using a pivot like $$ C \left( \mathbf{x} \right) = \left\{ \theta_{0} : a \le Q \left( \mathbf{x} ; \theta_{0} \right) \le b \right\} $$ is called pivoting.