Mixtures of increasing failure rate distributions can decrease, at least in some intervals of time. Usually this property is observed asymptotically, as t → ∞, which is due to the fact that a mixture failure rate is ‘bent down’, as the weakest populations are dying out first. We consider a survival model that generalizes additive hazards models, proportional hazards models, and accelerated life models very well known in reliability and survival analysis. We obtain new explicit asymptotic relations for a general setting and study specific cases. Under reasonable assumptions we prove that the asymptotic behavior of the mixture failure rate depends only on the behavior of the mixing distribution in the neighborhood of the left-hand endpoint of its support, and not on the whole mixing distribution.

Recently new partial orderings s-SFR, s-FR, s-ST, s-CV and s-CX have been defined in the literature. These are more general in the sense that most of the earlier known partial orderings reduce to particular cases of these orderings. Moreover, these orderings have helped in defining a new and useful ageing criterion. In this paper, we discuss preservation properties of these orderings under mixtures. Results are expected to be useful in engineering and reliability, where mixtures of distributions arise naturally in numerous situations. Moreover, existing results, specifically for LR, FR and ST orderings under the operation of mixtures, reduce to special cases.

A model incorporating the effect of a common environment on several components (structurally independent) of a system is developed. A multivariate generalization of the Lomax (Pareto type 2) distribution is obtained by mixing exponential variables. Its relationship to other multivariate distributions is discussed. Several properties of this distribution are reported and their usefulness in reliability theory indicated. Finally, a further generalization of this multivariate Lomax distribution is presented.

Any probability distribution on [0,∞) can function as the mixing distribution for a Poisson mixture, i.e. a mixture of Poisson distributions. The mixing distribution is called quasi-infinitely divisible (q.i.d.) if it renders the Poisson mixture infinitely divisible, or λ-q.i.d. if it does so after scaling by a factor λ> 0, or ∗-q.i.d. if it is λ-q.i.d. for some λ. These classes of distributions include the infinitely divisible distributions, and each exhibits many of the properties of the latter class but in weakened form. The paper presents the main properties of the classes and the class of Poisson mixtures, including characterisations of membership, relation with cumulants, and closure properties. Examples are given that establish among other things strict inclusions between the classes of mixing distributions.