Revealing the dependence structure between X(1) and X(n)

Abstract In this paper we address the dependence structure of the minimum and maximum of n iid random variables X 1 ,…, X n by determining their copula. It is then easy to give an alternative proof for their asymptotic independence and to calculate Kendall's τ and Spearman's ρ for ( X (1) , X ( n ) ). This will show that the dependence between the variables is already small for small sample sizes. Finally, it can be shown that 3 τ n ⩾ ρ n ⩾ τ n >0. Although closed-form expressions are available for τ n and ρ n , we cannot compare them directly but have to use the concept of positive likelihood ratio dependence to establish this result.