We study failure rate monotonicity and generalised convex transform stochastic ordering properties of random variables, with an emphasis on applications. We are especially interested in the effect of a tail-weight iteration procedure to define distributions, which is equivalent to the characterisation of moments of the residual lifetime at a given instant. For the monotonicity properties, we are mainly concerned with hereditary properties with respect to the iteration procedure providing counterexamples showing either that the hereditary property does not hold or that inverse implications are not true. For the stochastic ordering, we introduce a new criterion, based on the analysis of the sign variation of a suitable function. This criterion is then applied to prove ageing properties of parallel systems formed with components that have exponentially distributed lifetimes.

The signature representation shows that the reliability of the system is a mixture of the reliability functions of the k-out-of-n systems. The first representation was obtained for systems with independent and identically distributed (IID) components and after it was extended to exchangeable (EXC) components. The purpose of the present paper is to extend it to the class of systems with identically distributed (ID) components which have a diagonal-dependent copula. We prove that this class is much larger than the class with EXC components. This extension is used to compare systems with non-EXC components.

Two definitions of Birnbaum’s importance measure for coherent systems are studied in the case of exchangeable components. Representations of these measures in terms of distribution functions of the ordered component lifetimes are given. As an example, coherent systems with failure-dependent component lifetimes based on the notion of sequential order statistics are considered. Furthermore, it is shown that the two measures are ordered in the case of associated component lifetimes. Finally, the limiting behavior of the measures with respect to time is examined.

Large samples from a light-tailed distribution often have a well-defined shape. This paper examines the implications of the assumption that there is a limit shape. We show that the limit shape determines the upper quantiles for a large class of random variables. These variables may be described loosely as continuous homogeneous functionals of the underlying random vector. They play an important role in evaluating risk in a multivariate setting. The paper also looks at various coefficients of tail dependence and at the distribution of the scaled sample points for large samples. The paper assumes convergence in probability rather than almost sure convergence. This results in an elegant theory. In particular, there is a simple characterization of domains of attraction.

Relative ageing describes how one system ages with respect to another. The ageing faster orders are used to compare the relative ageing of two systems. Here, we study ageing faster orders in the hazard and reversed hazard rates. We provide some sufficient conditions for one coherent system to dominate another with respect to ageing faster orders. Further, we investigate whether the active redundancy at the component level is more effective than that at the system level with respect to ageing faster orders, for a coherent system. Furthermore, a used coherent system and a coherent system made out of used components are compared with respect to ageing faster orders.

We consider coherent systems with independent and identically distributed components. While it is clear that the system’s life will be stochastically larger when the components are replaced with stochastically better components, we show that, in general, similar results may not hold for hazard rate, reverse hazard rate, and likelihood ratio orderings. We find sufficient conditions on the signature vector for these results to hold. These results are combined with other well-known results in the literature to get more general results for comparing two systems of the same size with different signature vectors and possibly with different independent and identically distributed component lifetimes. Some numerical examples are also provided to illustrate the theoretical results.

The main result of this note implies that any function from the product of several vector spaces to a vector space can be uniquely decomposed into the sum of mutually orthogonal functions that are odd in some of the arguments and even in the other arguments. Probabilistic notions and facts are employed to simplify statements and proofs.

This paper presents a flexible family which we call the $\alpha$-mixture of survival functions. This family includes the survival mixture, failure rate mixture, models that are stochastically closer to each of these conventional mixtures, and many other models. The $\alpha$-mixture is endowed by the stochastic order and uniquely possesses a mathematical property known in economics as the constant elasticity of substitution, which provides an interpretation for $\alpha$. We study failure rate properties of this family and establish closures under monotone failure rates of the mixture’s components. Examples include potential applications for comparing systems.

In a recent article, Döring et al. (2018) conditioned a stable process with two-sided jumps to avoid an interval. As usual, the strategy was to find an invariant function for the process killed on entering the interval and to show that the corresponding h-transformed process is indeed the process conditioned to avoid an interval in a meaningful way. In the present article we consider the case of a completely asymmetric stable process. It turns out that the invariant function found by Döring et al. does not exist or is not invariant, but nonetheless we will characterise the conditioned process as a Markov process.

An upper bound for the hazard rate function of a convolution of not necessarily independent random lifetimes is provided, which generalizes a recent result established for independent random lifetimes. Similar results are considered for the reversed hazard rate function. Applications to parametric and semiparametric models are also given.

In this paper, we introduce a new large family of Lévy-driven point processes with (and without) contagion, by generalising the classical self-exciting Hawkes process and doubly stochastic Poisson processes with non-Gaussian Lévy-driven Ornstein–Uhlenbeck-type intensities. The resulting framework may possess many desirable features such as skewness, leptokurtosis, mean-reverting dynamics, and more importantly, the ‘contagion’ or feedback effects, which could be very useful for modelling event arrivals in finance, economics, insurance, and many other fields. We characterise the distributional properties of this new class of point processes and develop an efficient sampling method for generating sample paths exactly. Our simulation scheme is mainly based on the distributional decomposition of the point process and its intensity process. Extensive numerical implementations and tests are reported to demonstrate the accuracy and effectiveness of our scheme. Moreover, we use portfolio risk management as an example to show the applicability and flexibility of our algorithms.

In this paper, we consider exponentiated location-scale model and obtain several ordering results between extreme order statistics in various senses. Under majorization type partial order-based conditions, the comparisons are established according to the usual stochastic order, hazard rate order and reversed hazard rate order. Multiple-outlier models are considered. When the number of components are equal, the results are obtained based on the ageing faster order in terms of the hazard rate and likelihood ratio orders. For unequal number of components, we develop comparisons according to the usual stochastic order, hazard rate order, and likelihood ratio order. Numerical examples are considered to illustrate the results.

One of the approaches to the Riemann Hypothesis is the Nyman–Beurling criterion. Cotangent sums play a significant role in this criterion. Here we investigate the values of these cotangent sums for various shifts of the argument.

We study the impact of a random environment on lifetimes of coherent systems with dependent components. There are two combined sources of this dependence. One results from the dependence of the components of the coherent system operating in a deterministic environment and the other is due to dependence of components of the system sharing the same random environment. We provide different sets of sufficient conditions for the corresponding stochastic comparisons and consider various scenarios, namely, (i) two different (as a specific case, identical) coherent systems operate in the same random environment; (ii) two coherent systems operate in two different random environments; (iii) one of the coherent systems operates in a random environment and the other in a deterministic environment. Some examples are given to illustrate the proposed reasoning.

In this paper, we analyse the set of all possible aggregate distributions of the sum of standard uniform random variables, a simply stated yet challenging problem in the literature of distributions with given margins. Our main results are obtained for two distinct cases. In the case of dimension two, we obtain four partial characterization results. For dimension greater than or equal to three, we obtain a full characterization of the set of aggregate distributions, which is the first complete characterization result of this type in the literature for any choice of continuous marginal distributions.

We establish that a random sum of independent and identically distributed (i.i.d.) random quantities has a log-concave cumulative distribution function (cdf) if (i) the random number of terms in the sum has a log-concave probability mass function (pmf) and (ii) the distribution of the i.i.d. terms has a non-increasing density function (when continuous) or a non-increasing pmf (when discrete). We illustrate the usefulness of this result using a standard actuarial risk model and a replacement model.

We apply this fundamental result to establish that a compound renewal process observed during a random time interval has a log-concave cdf if the observation time interval and the inter-renewal time distribution have log-concave densities, while the compounding distribution has a decreasing density or pmf. We use this second result to establish the optimality of a so-called (s,S) policy for various inventory models with a stock-out cost coefficient of dimension [$/unit], significantly generalizing the conditions for the demand and leadtime processes, in conjunction with the cost structure in these models. We also identify the implications of our results for various algorithmic approaches to compute optimal policy parameters.

This paper studies the variability of both series and parallel systems comprised of heterogeneous (and dependent) components. Sufficient conditions are established for the star and dispersive orderings between the lifetimes of parallel [series] systems consisting of dependent components having multiple-outlier proportional hazard rates and Archimedean [Archimedean survival] copulas. We also prove that, without any restriction on the scale parameters, the lifetime of a parallel or series system with independent heterogeneous scaled components is larger than that with independent homogeneous scaled components in the sense of the convex transform order. These results generalize some corresponding ones in the literature to the case of dependent scenarios or general settings of components lifetime distributions.

We characterize various forms of positive dependence, such as association, positive supermodular association and dependence, and positive orthant dependence, for jump-Feller processes. Such jump processes can be studied through their state-space dependent Lévy measures. It is through these Lévy measures that we will provide our characterization. Finally, we present applications of these results to stochastically monotone Feller processes, including Lévy processes, the Ornstein–Uhlenbeck process, pseudo-Poisson processes, and subordinated Feller processes.

In this paper we treat a two-stage grouping procedure of building a k-out-of-n system from several clusters of components. We use a static framework in which the component reliabilities are fixed. Under such a framework, we address the impact of the selecting strategies, the sampling probabilities, and the component reliabilities on the constructed system’s reliability. An interesting finding is that the level of component reliabilities could be identified as a decisive factor in determining how the selecting strategies and the component reliabilities affect the system reliability. The new results generalize and extend those established earlier in the literature such as Di Crescenzo and Pellerey (2011), Hazra and Nanda (2014), Navarro, Pellerey, and Di Crescenzo (2015), and Hazra, Finkelstein, and Cha (2017). Several Monte Carlo simulation experiments are provided to illustrate the theoretical results.