On Dependence Properties of Random Minima and Maxima

Let X _(n) and X ₍₁₎ be the largest and smallest order statistics, respectively, of a random sample of fixed size n. Quite generally, X ₍₁₎ and X _(n) are approximately independent for n sufficiently large. In this article, we study the dependence properties of random extremes in terms of their copula, when the sample size has a left-truncated binomial distribution and show that they tend to be more dependent in this case. We also give closed-form formulas for the measures of association Kendall's τ and Spearman's ρ to measure the amount of dependence between two extremes.