Study of some measures of dependence between order statistics and systems

Abstract

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 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 $X_1,X_2,\dots ,X_n$ are independent and identically distributed or when they jointly have an exchangeable distribution.