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.

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, are a frequentist method to construction prediction intervals, and the use of pivotal quantities improves performance of the bootstrap.

Example 1

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 $X_{1},\ldots ,X_{n}$.

Example 2

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)