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 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 X 1 , X 2 , … , X n are independent and identically distributed or when they jointly have an exchangeable distribution.