ABSTRACT
A special example of an order statistic was introduced in Chapter 9 of Book II on extreme value theory, where this random variable was defined as the maximum of a collection of independent, identically distributed random variables. Order statistics generalize this notion, converting such a collection into ordered random variables. Distribution and density functions of such variates are first derived, before turning to the joint distribution function of all order statistics. This latter derivation will introduce some needed combinatorial ideas, as well as results on multivariate integration from Book III and a more general result from Book V. Various density functions of order statistics are then derived beginning with the joint density, and then on to various marginal and conditional densities of these random variables. The final investigation is the Rényi representation theorem for the order statistics from an exponential distribution. Remarkably, this result proves that the gaps between such ordered variates are independent exponentials with well-defined parameters. While seemingly of narrow applicability as a result on exponential variables, this theorem will be seen to be more widely applicable because exponential variables are readily transformed to and from uniform continuous variables. As developed in Book II, uniform continuous variables are fundamental to general random variable generation using a left-continuous inverse function F*(y).
