We compare distributions of residual lifetimes of dependent components of different age. This approach yields several notions of multivariate ageing. A special feature of our notions is that they are based on one-dimensional stochastic comparisons. Another difference from the traditional approach is that we do not condition on different histories.

Simple analytical lower and upper bounds are obtained for stop-loss premiums and ruin probabilities of compound Poisson risks in case the mean, variance and range of the claim size distribution are known. They are based on stop-loss extremal distributions and improve the bounds derived earlier from dangerous extremal distributions. The special bounds obtained in case the relative variance of the claim size is unknown, but its maximal value is known, are related to other actuarial results.

The paper is first concerned with a comparison of the partial sums associated with two sequences of n exchangeable Bernoulli random variables. It then considers a situation where such partial sums are obtained through an iterative procedure of branching type stopped at the first-passage time in a linearly decreasing upper barrier. These comparison results are illustrated with applications to certain urn models, sampling schemes and epidemic processes. A key tool is a non-standard hierarchical class of stochastic orderings between discrete random variables valued in {0, 1,· ··, n}.

A new partial ordering among life distributions in terms of their uncertainties is introduced. Our measure of uncertainty is Shannon information applied to the residual lifetime. The relationship between this ordering and various existing orderings of life distributions are discussed. Various properties of our proposed concept are examined. Based on our proposed ordering and various existing orderings, the notion of a ‘better system' is introduced.

Two devices are subjected to common shocks arriving according to two identical counting processes. Let and denote the probability of surviving k shocks for the first and the second device, respectively. We find conditions on the discrete distributions and in order to obtain the failure rate order (FR), the likelihood ratio order (LR) and the mean residual order (MR) between the random lifetimes of the two devices. We also obtain sufficient conditions under which the above mentioned relations between the discrete distributions are verified in some cumulative damage shock models.

Motivated by the need of studying a subset of components, ‘separate' from the other components, we introduce a new definition of ‘marginal distribution'. This is done by fixing the lives of the other components, but without the ‘knowledge' of the components of interest. Formally this is done by minimally repairing the components of no interest up to a predetermined time. Preservation properties of these ‘conditional marginal distributions', with respect to several stochastic orderings, are obtained. Also, inheritance of positive dependence properties, by the conditional marginal distributions, is shown. In addition, the preservations of dynamic multivariate aging properties, by the dynamic conditional marginal distributions, are obtained. The definitions and results are illustrated by a set of examples. Some applications for modelling ‘combinations' of sets of random lifetimes, and for bounding complex sets of random lifetimes, are described.