Diagonal sections of copulas, multivariate conditional hazard rates and distributions of order statistics for minimally stable lifetimes

Abstract

As a motivating problem, we aim to study some special aspects of the marginal distributions of the order statistics for exchangeable and (more generally) for minimally stable non-negative random variables $T_{1},...,T_{r}$. In any case, we assume that $T_{1},...,T_{r}$ are identically distributed, with a common survival function $\overline{G}$ and their survival copula is denoted by $K$. The diagonal's and subdiagonals' sections of $K$, along with $\overline{G}$, are possible tools to describe the information needed to recover the laws of order statistics. When attention is restricted to the absolutely continuous case, such a joint distribution can be described in terms of the associated multivariate conditional hazard rate (m.c.h.r.) functions. We then study the distributions of the order statistics of $T_{1},...,T_{r}$ also in terms of the system of the m.c.h.r. functions. We compare and, in a sense, we combine the two different approaches in order to obtain different detailed formulas and to analyze some probabilistic aspects for the distributions of interest. This study also leads us to compare the two cases of exchangeable and minimally stable variables both in terms of copulas and of m.c.h.r. functions. The paper concludes with the analysis of two remarkable special cases of stochastic dependence, namely Archimedean copulas and load sharing models. This analysis will allow us to provide some illustrative examples, and some discussion about peculiar aspects of our results.