In this paper we investigate the characterizations of life distributions under four stochastic orderings, < p, < (p), < (p) and < L, by a unified method. Conditions for the stochastic equality of two non-negative random variables under the four stochastic orderings are derived. Many previous results are consequences. As applications, we provide characterizations of life distributions by a single value of their Laplace transforms under orderings < p and < (p) and their moment generating functions under orderings < p and < (p). Under ordering < L, a characterization is given by the expected value of a strictly completely monotone function. The conditions for the stochastic equality of two non-negative vectors under the stochastic orderings < (p), < (p) and < L are presented in terms of the Laplace transforms and moment generating functions of their extremes and sample means. Characterizations of the exponential distribution among L and L life distribution classes are also given.

Vinogradov (1973) used the Laplace transform to characterize the IFR class of life distributions and later Block and Savits (1980) extended the characterization to the main reliability classes. Here we use the same transform again to characterize the continuous time renewal equation and some properties of its solution.

The accuracy of the Normal or Poisson approximations can be significantly improved by adding part of an asymptotic expansion in the exponent. The signed-compound-Poisson measures obtained in this manner can be of the same structure as the Poisson distribution. For large deviations we prove that signed-compound-Poisson measures enlarge the zone of equivalence for tails.

In the series system (competing risks) set-up the observed data are generally accepted as the lifetime (T) and the identifier (δ) of the component causing the failure of the system. Peterson (1976) has provided bounds for the joint survival function of the component lifetimes in terms of the joint distribution of (T, δ). In the case of more complex coherent systems, there are various schemes of observation in the literature. In this paper we provide bounds for the joint and marginal survival functions of the component lifetimes in terms of the joint distribution of the data as obtained under existing and new schemes of observation. We also tackle the reverse problem of obtaining bounds for the joint distributions of the data for given marginal distributions of the component lifetimes and the distribution of the system lifetimes.

We consider probability metrics of the following type: for a class of functions and probability measures P, Q we define A unified study of such integral probability metrics is given. We characterize the maximal class of functions that generates such a metric. Further, we show how some interesting properties of these probability metrics arise directly from conditions on the generating class of functions. The results are illustrated by several examples, including the Kolmogorov metric, the Dudley metric and the stop-loss metric.

Some moment inequalities are known to be valid for non-parametric lifetime distribution classes. Here we consider one set of these inequalities, which hold for random variables that are DMRL (decreasing in mean residual life). We prove that such inequalities are satisfied by variables which are sums of DMRL random variables too, though these sums are not necessarily DMRL. Related results are shown, together with similar results valid for the stochastic comparison in mean residual life.

Let X1, X2,… be a sequence of independent random variables and let N be a positive integer-valued random variable which is independent of the Xi. In this paper we obtain some stochastic comparison results involving min {X1, X2,…, XN) and max{X1, X2,…, XN}.

The purpose of this paper is to study two notions of stochastic comparisons of non-negative random variables via ratios that are determined by their Laplace transforms. Some interpretations of the new orders are given, and various properties of them are derived. The relationships to other stochastic orders are also studied. Finally, some applications in reliability theory are described.

The sojourn time that a Markov chain spends in a subset E of its state space has a distribution that depends on the hitting distribution on E and the probabilities (resp. rates in the continuous-time case) that govern the transitions within E. In this note we characterise the set of all hitting distributions for which the sojourn time distribution is geometric (resp. exponential).

We consider stochastic orders of the following type. Let be a class of functions and let P and Q be probability measures. Then define , if ∫ ⨍ d P ≦ ∫ ⨍ d Q for all f in . Marshall (1991) posed the problem of characterizing the maximal cone of functions generating such an ordering. We solve this problem by using methods from functional analysis. Another purpose of this paper is to derive properties of such integral stochastic orders from conditions satisfied by the generating class of functions. The results are illustrated by several examples. Moreover, we show that the likelihood ratio order is closed with respect to weak convergence, though it is not generated by integrals.

The Mellin-Stieltjes convolution and related decomposition of distributions in M(α) (the class of distributions μ on (0, ∞) with slowly varying αth truncated moments ) are investigated. Maller shows that if X and Y are independent non-negative random variables with distributions μ and v, respectively, and both μ and v are in D2, the domain attraction of Gaussian distribution, then the distribution of the product XY (that is, the Mellin-Stieltjes convolution μ ^ v of μ and v) also belongs to it. He conjectures that, conversely, if μ ∘ v belongs to D2, then both μ and v are in it. It is shown that this conjecture is not true: there exist distributions μ ∈ D2 and v μ ∈ D2 such that μ ^ v belongs to D2. Some subclasses of D2 are given with the property that if μ ^ v belongs to it, then both μ and v are in D2.

This paper establishes a compound Poisson limit theorem for the sum of a sequence of multi-state Markov chains. Our theorem generalizes an earlier one by Koopman for the two-state Markov chain. Moreover, a similar approach is used to derive a limit theorem for the sum of the k th-order two-state Markov chain.

A characterization of the exponential distribution based on a relevation-type equation and its discrete version are extended to the case of multidimensional spaces via variants of the Choquet–Deny theorem. Comments on some recent results in the literature are made.

The theory of majorization is applied to investigate the effects of heterogeneity in a class of epidemics. In particular, the heterogeneity of the components of the contact vector of a given susceptible is studied. It is shown that if the heterogeneity can be partially ordered through majorization, then the probability functions of escaping infection can be similarly ordered. The results can be applied to AIDS research on the spread of the HIV virus.

The shape of a rectangular prism in (d + 1)-dimensions is defined as Y = (Y1, Y2, · ··, Yd), Yn = Ln/Ln+1 where the Ln are the prism's edge lengths, in ascending order. We investigate shape distributions that are invariant when the prism is cut into two, also rectangular, prisms, with one prism retained for measurement and the other discarded. Interesting new distributions on [0, 1]d arise.

In this article we define the s-order equilibrium distribution of a life distribution and establish some interesting properties regarding moments for s-order equilibrium distributions. Results are expected to be useful in reliability and renewal processes.

An aggregated Markov chain is a Markov chain for which some states cannot be distinguished from each other by the observer. In this paper we consider the identifiability problem for such processes in continuous time, i.e. the problem of determining whether two parameters induce identical laws for the observable process or not. We also study the order of a continuous-time aggregated Markov chain, which is the minimum number of states needed to represent it. In particular, we give a lower bound on the order. As a by-product, we obtain results of this kind also for Markov-modulated Poisson processes, i.e. doubly stochastic Poisson processes whose intensities are directed by continuous-time Markov chains, and phase-type distributions, which are hitting times in finite-state Markov chains.

The Stein–Chen method for Poisson approximation is adapted to the setting of the geometric distribution. This yields a convenient method for assessing the accuracy of the geometric approximation to the distribution of the number of failures preceding the first success in dependent trials. The results are applied to approximating waiting time distributions for patterns in coin tossing, and to approximating the distribution of the time when a stationary Markov chain first visits a rare set of states. The error bounds obtained are sharper than those obtainable using related Poisson approximations.

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}.

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.