Study of some measures of dependence between order statistics and systems

Let be a random vector, and denote by X_1:n,X_2:n,…,X_n:n the corresponding order statistics. When X₁,X₂,…,X_n represent the lifetimes of n components in a system, the order statistic X_n−k+1:n represents the lifetime of a k-out-of-n system (i.e., a system which works when at least k components work). In this paper, we obtain some expressions for the Pearson’s correlation coefficient between X_i:n and X_j:n. We pay special attention to the case n=2, that is, to measure the dependence between the first and second failure in a two-component parallel system. We also obtain the Spearman’s rho and Kendall’s tau coefficients when the variables X₁,X₂,…,X_n are independent and identically distributed or when they jointly have an exchangeable distribution.