In this work, we consider two sets of dependent variables $\{X_{1},\ldots,X_{n}\}$ and $\{Y_{1},\ldots,Y_{n}\}$, where $X_{i}\sim EW(\alpha_{i},\lambda_{i},k_{i})$ and $Y_{i}\sim EW(\beta_{i},\mu_{i},l_{i})$, for $i=1,\ldots, n$, which are coupled by Archimedean copulas having different generators. We then establish different inequalities between two extremes, namely, $X_{1:n}$ and $Y_{1:n}$ and $X_{n:n}$ and $Y_{n:n}$, in terms of the usual stochastic, star, Lorenz, hazard rate, reversed hazard rate and dispersive orders. Several examples and counterexamples are presented for illustrating all the results established here. Some of the results here extend the existing results of [5] (Barmalzan, G., Ayat, S.M., Balakrishnan, N., & Roozegar, R. (2020). Stochastic comparisons of series and parallel systems with dependent heterogeneous extended exponential components under Archimedean copula. Journal of Computational and Applied Mathematics 380: Article No. 112965).

In this paper, the ordering properties of convex and increasing convex orders of the dependent random variables are studied. Some closure properties of the convex and increasing convex orders under independent random variables are extended to the dependent random variables under the Archimedean copula. Two applications are provided to illustrate our results.

In the literature on active redundancy allocation, the redundancy lifetimes are usually postulated to be independent of the component lifetimes for the sake of technical convenience. However, this unrealistic assumption leads to a risk of inaccurately evaluating system reliability, because it overlooks the statistical dependence of lifetimes due to common stresses. In this study, for k-out-of-n:F systems with component and redundancy lifetimes linked by the Archimedean copula, we show that (i) allocating more homogeneous redundancies to the less reliable components tends to produce a redundant system with stochastically larger lifetime, (ii) the reliability of the redundant system can be uniformly maximized through balancing the allocation of homogeneous redundancies in the context of homogeneous components, and (iii) allocating a more reliable matched redundancy to a less reliable component produces a more reliable system. These novel results on k-out-of-n:F systems in which component and redundancy lifetimes are statistically dependent are more applicable to the complicated engineering systems that arise in real practice. Some numerical examples are also presented to illustrate these findings.

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.

Catastrophe insurance markets fail to provide sufficient protections against natural catastrophes, whereas they have the capacity to absorb the losses. In this paper, we assume the catastrophic risks are dependent and extremely heavy-tailed, and insurers have limited liability to cover losses up to a certain amount. We provide a comprehensive study to show that the diversification in the catastrophe insurance markets can be transited from suboptimal to preferred by increasing the number of insurers in the market. This highlights the importance of coordination among insurers and the government intervention in encouraging insurers to participate in the catastrophe insurance market to exploit risk sharing. Simulation studies are provided to illuminate the key findings of our results.

In this paper, we discuss stochastic orderings of lifetimes of two heterogeneous parallel and series systems with heterogeneous dependent components having generalized Birnbaum–Saunders distributions. The comparisons presented here are based on the vector majorization of parameters. The ordering results are established in some special cases for the generalized Birnbaum–Saunders distribution based on the multivariate elliptical, normal, t, logistic, and skew-normal kernels. Further, we use these results by considering Archimedean copulas to model the dependence structure among systems with generalized Birnbaum–Saunders components. These results have been used to derive some upper and lower bounds for survival functions of lifetimes of parallel and series systems.

Consider a random vector $\textbf{U}$ whose distribution function coincides in its upper tail with that of an Archimedean copula. We report the fact that the conditional distribution of $\textbf{U}$ , conditional on one of its components, has under a mild condition on the generator function independent upper tails, no matter what the unconditional tail behavior is. This finding is extended to Archimax copulas.

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.

Considering the Gaussian noise channel, Costa [4] investigated the concavity of the entropy power when the input signal and noise components are independent. His argument was connected to the first-order derivative of the Fisher information. In real situations, however, the noise can be highly dependent on the main signal. In this paper, we suppose that the input signal and noise variables are dependent. Then, some well-known copula functions are used to define their dependence structure. The first- and second-order derivatives of Fisher information of the model are obtained. Then, by using these derivatives, we will generalize two inequalities based on the Fisher information and a functional that is closely associated to Fisher information for the case when the input signal and noise variables are dependent. We will also show that the previous results for the independent case are recovered as special cases of our result. Several applications are provided to support the usefulness of our results. Finally, the channel capacity of the Gaussian noise channel model with dependent signal and noise is studied.

Conditional Monte Carlo replaces a naive estimate Z of a number z by its conditional expectation given a suitable piece of information. It always reduces variance and its traditional applications are in that vein. We survey here other potential uses such as density estimation and calculations for Value-at-Risk and/or expected shortfall, going in part into the implementation in various copula structures. Also the interplay between these different aspects comes into play.

In this paper we build the increasing convex (concave) order for the scalar product of random vectors with an upper (lower) tail permutation decreasing joint density. As applications, we revisit allocations of portfolio risks in financial engineering and of coverage limits and deductibles in insurance. Some related results in the literature are substantially updated.

In this paper we develop two permutation theorems on argument increasing functions of a multivariate random vector and a real parameter vector. We use the unified approach of our two theorems to provide some important theoretical results on the capital allocation in actuarial science, the deductible and upper limit allocations in insurance policy, and portfolio allocation in financial engineering. Our results successfully improve or extend the corresponding works in the literature.

We consider the multivariate Value-at-Risk (VaR) and Conditional-Tail-Expectation (CTE) risk measures introduced in Cousin and Di Bernardino (Cousin, A. and Di Bernardino, E. (2013) Journal of Multivariate Analysis, 119, 32–46; Cousin, A. and Di Bernardino, E. (2014) Insurance: Mathematics and Economics, 55(C), 272–282). For absolutely continuous Archimedean copulas, we derive integral formulas for the multivariate VaR and CTE Archimedean risk measures. We show that each component of the multivariate VaR and CTE functional vectors is an integral transform of the corresponding univariate VaR measures. For the class of Archimedean copulas, the marginal components of the CTE vector satisfy the following properties: positive homogeneity (PH), translation invariance (TI), monotonicity (MO), safety loading (SL) and VaR inequality (VIA). In case marginal risks satisfy the subadditivity (MSA) property, the marginal CTE components are also sub-additive and hitherto coherent risk measures in the usual sense. Moreover, the increasing risk (IR) or stop-loss order preserving property of the marginal CTE components holds for the class of bivariate Archimedean copulas. A counterexample to the (IR) property for the trivariate Clayton copula is included.

Tail comonotonicity, or asymptotic full dependence, is proposed as a reasonable conservative dependence structure for modeling dependent risks. Some sufficient conditions have been obtained to justify the conservativity of tail comonotonicity. Simulation studies also suggest that, by using tail comonotonicity, one does not lose too much accuracy but gain reasonable conservative risk measures, especially when considering high scenario risks. A copula model with tail comonotonicity is applied to an auto insurance dataset. Particular models for tail comonotonicity for loss data can be based on the BB2 and BB3 copula families and their multivariate extensions.

This paper covers a number of different topics related to the measurement and modelling of dependency within economic capital models. The scope of the paper is relatively wide. We address in some detail the different approaches to modelling dependencies ranging from the more common variance-covariance matrix approach, to the consideration of the use of copulas and the more sophisticated causal models that feature feedback loops and other systems design ideas.

There are many data and model uncertainties in modelling dependency and so we have also endeavoured to cover topics such as spurious relationships and wrong-way risk to highlight some of the uncertainties involved.

With the advent of the internal model approval process under Solvency II, senior management needs to have a greater understanding of dependency methodology. We have devoted a section of this paper to a discussion of possible different ways to communicate the results of modelling to the board, senior management and other interested parties within an insurance company.

We have endeavoured throughout this paper to include as many numerical examples as possible to help in the understanding of the key points, including our discussion of model parameterisation and the communication to an insurance executive of the impact of dependency on economic capital modelling results.

The economic capital model can be seen as a combination of two key components: the marginal risk distribution of each risk and the aggregation methodology which combines these into a single aggregate distribution or capital number. This paper is concerned with the aggregation part, the methods and assumptions employed and the issues arising, and not the determination of the marginal risk distributions which is equally of importance and in many cases equally as complex.

We show how the extremal behavior of d-variate Archimedean copulas can be deduced from their stochastic representation as the survival dependence structure of an ℓ1-symmetric distribution (see McNeil and Nešlehová (2009)). We show that the extremal behavior of the radial part of the representation is determined by its Williamson d-transform. This leads in turn to simple proofs and extensions of recent results characterizing the domain of attraction of Archimedean copulas, their upper and lower tail-dependence indices, as well as their associated threshold copulas. We outline some of the practical implications of their results for the construction of Archimedean models with specific tail behavior and give counterexamples of Archimedean copulas whose coefficient of lower tail dependence does not exist.

We consider a dependent portfolio of insurance contracts. Asymptotic tail probabilities of the ECOMOR and LCR reinsurance amounts are obtained under certain assumptions about the dependence structure.

The tail behavior of sums of dependent risks was considered by Wüthrich (2003) and by Alink et al. (2004, 2005) in the case where the variables are exchangeable and connected through an Archimedean copula model. It is shown here how their result can be extended to a broader class of dependence structures using multivariate extreme-value theory. An explicit form is given for the asymptotic probability of extremal events, and the behavior of the latter is studied as a function of the indices of regular variation of both the copula and the common distribution of the risks.

We consider d identically and continuously distributed dependent risks X 1,…, Xd . Our main result is a theorem on the asymptotic behaviour of expected shortfall for the aggregate risks: there is a constant cd such that for large u we have . Moreover we study diversification effects in two dimensions, similar to our Value-at-Risk studies in [2].

We estimate Value-at-Risk for sums of dependent random variables. We model multivariate dependent random variables using archimedean copulas. This structure allows one to calculate the asymptotic behaviour of extremal events. An important application of such results are Value-at-Risk estimates for sums of dependent random variables.