On total positivity of exchangeable random variables obtained by symmetrization, with applications to failure-dependent lifetimes

Abstract

Necessary and sufficient conditions for multivariate total positivity of order 2 (MTP2) for density functions of some class of exchangeable random variables are obtained. The considered densities occur via symmetrization of particular ordered random variables. As an example, a characterization of the MTP2 property for the Freund–Weinman multivariate exponential distribution is given. Furthermore, the results are applied to general failure-dependent component lifetimes in systems based on sequential order statistics. In the latter setting, the hazard rate increasing upon failure (HIF) property is also characterized. In particular, the case of underlying distributions satisfying the proportional hazards assumption is considered. The results are supplemented by an analysis of the covariances of the above multivariate exponential distribution.