Pivotal quantity

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In statistics, a pivotal quantity is a function of observations whose distribution does not depend on unknown parameters.

More formally, given an independent and identically distributed sample $X = (X_1,X_2,\ldots,X_n)$ from a distribution with parameter $θ$ , a function $g$ is a pivotal quantity if the distribution of $g (X,θ)$ is independent of $θ$ .

It is relatively easy to construct pivots for location and scale parameters: for the former we form differences, for the latter ratios.

Pivotal quantities provide one method of constructing confidence intervals.

[edit] Example

Given $n$ independent, identically distributed (i.i.d.) observations $X = (X_1, X_2, \ldots, X_n)$ from the normal distribution with unknown mean $μ$ and variance $σ 2$ , a pivotal quantity can be obtained from the function:

$g(x,X) = \frac{x - \overline{X}}{s}$

where

$\overline{X} = \frac{1}{n}\sum_{i=1}^n{X_i}$

and

$s^2 = \frac{1}{n-1}\sum_{i=1}^n{(X_i - \overline{X})^2}$

are unbiased estimates of $μ$ and $σ 2$ , respectively. The function $g (x, X)$ is the Student's t-statistic for a new value $x$ , to be drawn from the same population as the already observed set of values $X$ .

Using $x = μ$ the function $g (μ, X)$ becomes a pivotal quantity, which is also distributed by the Student's t-distribution with $ν = n - 1$ degrees of freedom. As required, even though $μ$ appears as an argument to the function $g$ , the distribution of $g (μ, X)$ does not depend on the parameters $μ$ or $σ$ of the normal probability distribution that governs the observations $X_1,\ldots,X_n$ .