Chapter 1
ORDER STATISTICS AND QUANTILES

Almost everything in this course is built from one operation: sorting the sample. Ordering the observations destroys the labels — which subject supplied which value — and keeps only the values and their positions. That loss is deliberate, and it is what buys the distribution-free property established in Chapter 2, since a statistic depending on the sample only through its ordering inherits the symmetry of the underlying variables.

Before that can be exploited we need the distributions of the ordered values themselves, and they are not the distributions of the original observations. The observations are independent; the order statistics are not, since \(X_{(1)}\leq X_{(2)}\leq \cdots \leq X_{(n)}\) by construction. The observations are identically distributed; the order statistics are not, the smallest and the largest having quite different distributions from the middle ones. This chapter derives those distributions — marginal, joint, and conditional — together with the moments that later chapters require.