Pivotal quantity

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 $heta$, a function $g$ is a pivotal quantity if the distribution of $g(X,\; heta)$ is independent of $heta$.

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 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 $u\; =\; n-1$ 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 $(X\_i,Y\_i)\text{'}$ is taken from bivariate normal distribution with unknown correlation $ho$. An estimator of $ho$ is the sample (Pearson, moment) correlation:$r\; =\; frac\{frac1\{n-1\}\; sum\_\{i=1\}^n\; (X\_i\; -\; overline\{X\})(Y\_i\; -\; overline\{Y\})\}\{s\_X\; s\_Y\}$where $s\_X,\; s\_Y$ are sample variances of $X$ and $Y$. Being a U-statistic, $r$ will have an asymptotically normal distribution::$sqrt\{n\}frac\{r-\; ho\}\{1-\; ho^2\}\; Rightarrow\; N(0,1)$.However, a variance stabilizing transformation:$z\; =\; m\{tanh\}^\{-1\}\; r\; =\; frac12\; ln\; frac\{1+r\}\{1-r\}$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 $zeta\; =\; \{\; m\; tanh\}^\{-1\}\; ho$ 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:$V\; [z]\; =\; frac1\{n-2\}$

References

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