Pivotal quantity

In statistics, a pivotal quantity is a function of observations whose distribution does not depend on unknown parameters. Note that a pivot quantity need not be a statistic – the function and its value can depend on parameters of the model, but its distribution must not. If it is a statistic, then it is known as an ancillary statistic.

More formally, given an independent and identically distributed sample $X=(X_{1},X_{2},\ldots ,X_{n})$ from a distribution with parameter $\theta$, a function $g$ is a pivotal quantity if the distribution of $g(X,\theta )$ is independent of $\theta$.

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, 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).

Examples

Normal distribution

See also: Prediction interval § Normal distribution

Given $n$ independent, identically distributed (i.i.d.) observations $X=(X_{1},X_{2},\ldots ,X_{n})$ from the normal distribution with unknown mean $\mu$ and variance $\sigma ^{2}$, 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 $\mu$ and $\sigma ^{2}$, 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 $x=\mu$ the function $g(\mu ,X)$ becomes a pivotal quantity, which is also distributed by the Student's t-distribution with $\nu =n-1$ degrees of freedom. As required, even though $\mu$ appears as an argument to the function $g$, the distribution of $g(\mu ,X)$ does not depend on the parameters $\mu$ or $\sigma$ of the normal probability distribution that governs the observations $X_{1},\ldots ,X_{n}$.

This can be used to compute a prediction interval for the next observation $X_{n+1};$ see Prediction interval: Normal distribution.

Bivariate normal distribtion

In more complicated cases, it is impossible to construct exact pivots. However, having approximate pivots improves convergence to asymptotic normality.

Suppose a sample of size $n$ of vectors $(X_{i},Y_{i})'$ is taken from bivariate normal distribution with unknown correlation $\rho$.

An estimator of $\rho$ is the sample (Pearson, moment) correlation

$r={\frac {{\frac {1}{n-1}}\sum _{i=1}^{n}(X_{i}-{\overline {X}})(Y_{i}-{\overline {Y}})}{s_{X}s_{Y}}}$

where $s_{X},s_{Y}$ are sample variances of $X$ and $Y$. Being a U-statistic, $r$ will have an asymptotically normal distribution:

${\sqrt {n}}{\frac {r-\rho }{1-\rho ^{2}}}\Rightarrow N(0,1)$.

However, a variance stabilizing transformation

$z={\rm {{tanh}^{-1}r={\frac {1}{2}}\ln {\frac {1+r}{1-r}}}}$

known as Fisher's $z$ transformation of the correlation coefficient allows to make the distribution of $z$ asymptotically independent of unknown parameters:

${\sqrt {n}}(z-\zeta )\Rightarrow N(0,1)$

where $\zeta ={\rm {tanh}}^{-1}\rho$ is the corresponding population parameter. For finite samples sizes $n$, the random variable $z$ will have distribution closer to normal than that of $r$. Even closer approximation to normality will be achieved by using the exact variance

$V[z]={\frac {1}{n-2}}$

References

Shao, J (2003) Mathematical Statistics, Springer, New York. ISBN 978-0-387-95382-3 (Section 7.1)