Introduction

Let denote a continuous random variable with probability density function (pdf) and cumulative distribution function (cdf) , and let denote a random sample of size drawn on . Let denote the random sample ordered such that ; then are collectively known as the order statistics derived from the parent . For example, is the smallest order statistic and corresponds to the sample minimum, and is the largest order statistic and corresponds to the sample maximum.

For a detailed discussion of the statistical theory pertaining to order statistics see, for example, [3, 4, 5, 6]. For the case in which is a discrete random variable, see [7]. On the computational side, Rose and Smith [2, Section 9.4] use mathStatica to obtain the algebraic and numeric properties of order statistics derived from a continuous parent. Whereas Evans et al. [8] appear to be restricted to numeric-only calculations, they also consider settings in which the parent variable is sampled without replacement.

Section 1 illustrates briefly the case of a continuous parent, while Sections 2 and 3 extend to the case of a discrete parent. Section 4 relaxes the independent and identically distributed (iid) assumptions, thereby illustrating some of the new functionality of mathStatica.

We begin by loading mathStatica 1.5 or later.