Study of some measures of dependence between order statistics and systems

Let be a random vector, and denote by X1:n,X2:n,...,Xn:n the corresponding order statistics. When X1,X2,...,Xn represent the lifetimes of n components in a system, the order statistic Xn-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 Xi:n and Xj: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,...,Xn are independent and identically distributed or when they jointly have an exchangeable distribution.