Study of some measures of dependence between order statistics and systems

Let X=(X1,X2,…,Xn) be a random vector, and denote by X1:n,X2:n,…X1:n,X2:n,…,Xn:nXn:n the corresponding order statistics. When X1,X2,…,XnX1,X2,…,Xn represent the lifetimes of nn components in a system, the order statistic Xn−k+1:nXn−k+1:n represents the lifetime of a kk-out-of-nn system (i.e., a system which works when at least kk components work). In this paper, we obtain some expressions for the Pearson’s correlation coefficient between Xi:nXi:n and Xj:nXj:n. We pay special attention to the case n=2n=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,…,XnX1,X2,…,Xn are independent and identically distributed or when they jointly have an exchangeable distribution.