Study of some measures of dependence between order statistics and systems
Let \(X=(X_1,X_2,\dots,X_n)\) be a random vector and denote by \(X_{1:n},X_{2:n},\dots,X_{n:n}\) the corresponding order statistics. When \(X_1,X_2,\dots,X_n\) represent the life times of \(n\) components in a system, the order statistic \(X_{n-k+1:n}\) represents the life time of a \(k\)-out-of-\(n\) system (i.e., a system which works when at least \(k\) components work). The authors obtain some expressions for the Pearson's correlation coefficient between \(X_{i:n}\) and \(X_{j:n}\). Special attention is paid to the case \(n=2\), that is, to measure the dependence between the first and second failure in a two-component parallel system. Spearman's rho and Kendall's tau coefficients are also obtained when \(X_1,X_2,\dots,X_n\) are independent and identically distributed or when they jointly have an exchangeable distribution.
scientific article; zbMATH DE number 5639704
scientific article