Chapter 2
DISTRIBUTION-FREE STATISTIC

Chapter 1 studied the order statistics themselves. This chapter asks what can be built from them, and in particular how a statistic can have a null distribution that is known even though the distribution generating the data is not.

The idea is best approached by noticing that it is not unique to non-parametric work. The \(t\)-statistic has a distribution free of \(\sigma ^{2}\), and the sample variance scaled by \(\sigma ^{2}_0\) has a distribution free of \(\mu \); in each case a nuisance parameter has cancelled. What follows generalises that cancellation. We define distribution-freeness relative to a class of distributions, observe that the classical statistics are distribution-free only over parametric classes, and then show — via the probability integral transform — that rank statistics achieve it over the class of all continuous distributions. That is the sense in which the subject is non-parametric, and Section 2.2 makes the distinction precise.