Study of some measures of dependence between order statistics and systems

Let X = (X-1, X-2,...,X-n) be a random vector, and denote by X-1:n, X-2:n,...,X-n:n the corresponding order statistics. When X-1, X-2,...,X-n 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 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 X-1, X-2,...,X-n are independent and identically distributed or when they jointly have an exchangeable distribution. (C) 2009 Elsevier Inc. All rights reserved.