Signature theory plays an important part in the field of reliability. In this paper, the ordered multi-state system signature and its related properties are discussed based on a life-test of independent and non-identical coherent or mixed systems with independent and identical binary-state components. Dynamic properties of these systems are considered through a new notion called dynamic multi-state system signature, and then related comparisons are made based on system lifetimes and costs. Finally, the theoretical results established are illustrated with some specific examples to demonstrate the use of dynamic ordered multi-state system signature in evaluating used multi-state coherent or mixed systems.

In this paper, we study the pricing of vulnerable Asian options with liquidity risk. We employ general Lévy processes to capture the changes in the liquidity discount factors and the information processes of all assets. In the proposed pricing model, we obtain the closed-form pricing formula of vulnerable Asian options using the Fourier transform methods. Finally, the derived pricing formula is used to illustrate the effects of asymmetric jump risk, and the effects are relatively stable on (vulnerable) Asian options with different moneynesses.

Copula is one of the widely used techniques to describe the dependency structure between components of a system. Among all existing copulas, the family of Archimedean copulas is the popular one due to its wide range of capturing the dependency structures. In this paper, we consider the systems that are formed by dependent and identically distributed components, where the dependency structures are described by Archimedean copulas. We study some stochastic comparisons results for series, parallel, and general $r$-out-of-$n$ systems. Furthermore, we investigate whether a system of used components performs better than a used system with respect to different stochastic orders. Furthermore, some aging properties of these systems have been studied. Finally, some numerical examples are given to illustrate the proposed results.

This paper studies the optimal allocation policy of a coherent system with independent heterogeneous components and dependent subsystems, the systems are assumed to consist of two groups of components whose lifetimes follow proportional hazard (PH) or proportional reversed hazard (PRH) models. We investigate the optimal allocation strategy by finding out the number $k$ of components coming from Group A in the up-series system. First, some sufficient conditions are provided in the sense of the usual stochastic order to compare the lifetimes of two-parallel–series systems with dependent subsystems, and we obtain the hazard rate and reversed hazard rate orders when two subsystems have independent lifetimes. Second, similar results are also obtained for two-series–parallel systems under certain conditions. Finally, we generalize the corresponding results to parallel–series and series–parallel systems with multiple subsystems in the viewpoint of the minimal path and the minimal cut sets, respectively. Some numerical examples are presented to illustrate the theoretical findings.

In this paper, we consider the problem of sustainable harvesting. We explain how the manager maximizes his/her profit according to the quantity of natural resource available in a harvesting area and under the constraint of penalties and fines when the quota is exceeded. We characterize the optimal values and some optimal strategies using a verification result. We then show by numerical examples that this optimal strategy is better than naive ones. Moreover, we define a level of fines which insures the double objective of the sustainable harvesting: a remaining quantity of available natural resource to insure its sustainability and an acceptable income for the manager.

Almost stochastic dominance has been receiving a great amount of attention in the financial and economic literatures. In this paper, we characterize the properties of almost first-order stochastic dominance (AFSD) via distorted expectations and investigate the conditions under which AFSD is preserved under a distortion transform. The main results are also applied to establish stochastic comparisons of order statistics and receiver operating characteristic curves via AFSD.

In this paper, the classical compound Poisson model under periodic observation is studied. Different from the random observation assumption widely used in the literature, we suppose that the inter-observation time is a constant. In this model, both the finite-time and infinite-time Gerber-Shiu functions are studied via the Laguerre series expansion method. We show that the expansion coefficients can be recursively determined and also analyze the approximation errors in detail. Numerical results for several claim size density functions are given to demonstrate effectiveness of our method, and the effect of some parameters is also studied.

We consider an extreme renewal process with no-mean heavy-tailed Pareto(II) inter-renewals and shape parameter $\alpha$ where $0\lt\alpha \leq 1$. Two steps are required to derive integral expressions for the analytic probability density functions (pdfs) of the fixed finite time $t$ excess, age, and total life, and require extensive computations. Step 1 creates and solves a Volterra integral equation of the second kind for the limiting pdf of a basic underlying regenerative process defined in the text, which is used for all three fixed finite time $t$ pdfs. Step 2 builds the aforementioned integral expressions based on the limiting pdf in the basic underlying regenerative process. The limiting pdfs of the fixed finite time $t$ pdfs as $t\rightarrow \infty$ do not exist. To reasonably observe the large $t$ pdfs in the extreme renewal process, we approximate them using the limiting pdfs having simple well-known formulas, in a companion renewal process where inter-renewals are right-truncated Pareto(II) variates with finite mean; this does not involve any computations. The distance between the approximating limiting pdfs and the analytic fixed finite time large $t$ pdfs is given by an $L_{1}$ metric taking values in $(0,1)$, where “near $0$” means “close” and “near $1$” means “far”.

We consider a collection of statistically identical two-state continuous time Markov chains (channels). A controller continuously selects a channel with the view of maximizing infinite horizon average reward. A switching cost is paid upon channel changes. We consider two cases: full observation (all channels observed simultaneously) and partial observation (only the current channel observed). We analyze the difference in performance between these cases for various policies. For the partial observation case with two channels or an infinite number of channels, we explicitly characterize an optimal threshold for two sensible policies which we name “call-gapping” and “cool-off.” Our results present a qualitative view on the interaction of the number of channels, the available information, and the switching costs.

In this paper, we discuss a generalization of the classical compound Poisson model with claim sizes following a compound distribution. As applications, we consider models involving zero-truncated geometric, zero-truncated negative-binomial and zero-truncated binomial batch-claim arrivals. We also provide some ruin-related quantities under the resulting risk models. Finally, through numerical examples, we visualize the behavior of these quantities.

The star-shaped ordering between probability distributions is a common way to express aging properties. A well-known criterion was proposed by Saunders and Moran [(1978). On the quantiles of the gamma and F distributions. Journal of Applied Probability 15(2): 426–432], to order families of distributions depending on one real parameter. However, the lifetime of complex systems usually depends on several parameters, especially when considering heterogeneous components. We extend the Saunders and Moran criterion characterizing the star-shaped order when the multidimensional parameter moves along a given direction. A few applications to the lifetime of complex models, namely parallel and series models assuming different individual components behavior, are discussed.

In this paper, we treat a nonlinear and unbalanced $2$-color urn scheme, subjected to two different nonlinear drawing rules, depending on the color withdrawn. We prove a central limit theorem as well as a law of large numbers for the urn composition. We also give an estimate of the mean and variance of both types of balls.

By now there is a solid theory for Polya urns. Finding the covariances is somewhat laborious. While these papers are “structural,” our purpose here is “computational.” We propose a practicable method for building the asymptotic covariance matrix in tenable balanced urn schemes, whereupon the asymptotic covariance matrix is obtained by solving a linear system of equations. We demonstrate the use of the method in growing tenable balanced irreducible schemes with a small index and in critical urns. In the critical case, the solution to the linear system of equations is explicit in terms of an eigenvector of the scheme.

This paper studies a generalization of the Gerber-Shiu expected discounted penalty function [Gerber and Shiu (1998). On the time value of ruin. North American Actuarial Journal 2(1): 48–72] in the context of the perturbed compound Poisson insurance risk model, where the moments of the total discounted claims and the discounted small fluctuations (arising from the Brownian motion) until ruin are also included. In particular, the latter quantity is represented by a stochastic integral and has never been analyzed in the literature to the best of our knowledge. Recursive integro-differential equations satisfied by our generalized Gerber-Shiu function are derived, and these are transformed to defective renewal equations where the components are identified. Explicit solutions are given when the individual claim amounts are distributed as a combination of exponentials. Numerical illustrations are provided, including the computation of the covariance between discounted claims and discounted perturbation until ruin.

For a bivariate random vector $(X, Y)$, suppose $X$ is some interesting loss variable and $Y$ is a benchmark variable. This paper proposes a new variability measure called the joint tail-Gini functional, which considers not only the tail event of benchmark variable $Y$, but also the tail information of $X$ itself. It can be viewed as a class of tail Gini-type variability measures, which also include the recently proposed tail-Gini functional. It is a challenging and interesting task to measure the tail variability of $X$ under some extreme scenarios of the variables by extending the Gini's methodology, and the two tail variability measures can serve such a purpose. We study the asymptotic behaviors of these tail Gini-type variability measures, including tail-Gini and joint tail-Gini functionals. The paper conducts this study under both tail dependent and tail independent cases, which are modeled by copulas with so-called tail order property. Some examples are also shown to illuminate our results. In particular, a generalization of the joint tail-Gini functional is considered to provide a more flexible version.

Let $X_1, \ldots, X_n$ be mutually independent exponential random variables with distinct hazard rates $\lambda _1, \ldots, \lambda _n$ and let $Y_1, \ldots, Y_n$ be a random sample from the exponential distribution with hazard rate $\bar \lambda = \sum _{i=1}^{n} \lambda _i/n$. Also let $X_{1:n} \lt \cdots \lt X_{n:n}$ and $Y_{1:n} \lt \cdots \lt Y_{n:n}$ be their associated order statistics. It is proved that for $1\le i \lt j \le n$, the generalized spacing $X_{j:n} - X_{i:n}$ is more dispersed than $Y_{j:n} - Y_{i:n}$ according to dispersive ordering and for $2\le i \le n$, the dependence of $X_{i:n}$ on $X_{1:n}$ is less than that of $Y_{i:n}$ on $Y_{1 :n}$, in the sense of the more stochastically increasing ordering. This dependence result is also extended to the proportional hazard rates (PHR) model. This extends the earlier work of Genest et al. [(2009)]. On the range of heterogeneous samples. Journal of Multivariate Analysis 100: 1587–1592] who proved this result for $i =n$.

In this paper, we identify some conditions to compare the largest order statistics from resilience-scale models with reduced scale parameters in the sense of mean residual life order. As an example of the established result, the exponentiated generalized gamma distribution is examined. Also, for the special case of the scale model, power-generalized Weibull and half-normal distributions are investigated.

This paper examines the preservation of several aging classes of lifetime distributions in the formation of coherent and mixed systems with independent and identically distributed (i.i.d.) or identically distributed (i.d.) component lifetimes. The increasing mean inactivity time class and the decreasing mean time to failure class are developed for the lifetime of systems with possibly dependent and i.d. component lifetimes. The increasing likelihood ratio property is also discussed for the lifetime of a coherent system with i.i.d. component lifetimes. We present sufficient conditions satisfied by the signature of a coherent system with i.i.d. components with exponential distribution, under which the decreasing mean remaining lifetime, the increasing mean inactivity time, and the decreasing mean time to failure are all satisfied by the lifetime of the system. Illustrative examples are presented to support the established results.

In this paper, we consider some dividend problems in the perturbed compound Poisson model under a constant barrier dividend strategy. We approximate the expected present value of dividend payments before ruin and the expected discounted penalty function based on the COS method, and construct some nonparametric estimators by using a random sample on claim number and individual claim sizes. Under a large sample size setting, we perform an error analysis of the estimators. We also provide some simulation results to verify the effectiveness of this estimation method when the sample size is finite.