Some new constructions of bivariate Weibull models

In this article, several approaches are advanced towards the construction of bivariate Weibull models from the consideration of failure behaviors of the components of a two-component system. First, a general method of construction of bivariate life models is developed in the setting of random environmental effects. Some new bivariate Weibull models are derived as special cases and added insights are provided for some of the existing ones. In the course of model formulation in terms of the dependence structure, a new bivariate family of life distributions is constructed so as to incorporate both positive and negative quadrant dependence in the same parametric setting, and a bivariate Weibull model is obtained as a special case. Finally, some distributional properties are presented for a bivariate Weibull model derived from the consideration of random hazards.     

Optimal capital allocations to interdependent actuarial risks

This paper further studies the capital allocation concerning mutually interdependent random risks. In the context of exchangeable random risks, we establish that risk-averse insurers incline to evenly distribute the total capital among multiple risks. For risk-averse insurers with decreasing convex loss functions, we prove that more capital should be allocated to the risk with the larger reversed hazard rate when risks are coupled by an Archimedean copula. Also, sufficient conditions are developed to exclude the worst capital allocations for random risks with some specific Archimedean copulas.     

Some General Characterizations of the Bivariate Gumbel Distribution and the Bivariate Lomax Distribution Based on Truncated Expectations

Recently attempts have been made to characterize probability distributions via truncated expectations in both univariate and multivariate cases. In this paper we will use a well known theorem of Lau and Rao (1982) to obtain some characterization results, based on the truncated expectations of a functionh, for the bivariate Gumbel distribution, a bivariate Lomax distribution, and a bivariate power distribution. The results of the paper subsume some earlier results appearing in the literature.     

Portfolio diversification under local and moderate deviations from power laws

This paper analyzes portfolio diversification for nonlinear transformations of heavy-tailed risks. It is shown that diversification of a portfolio of convex functions of heavy-tailed risks increases the portfolio’s riskiness if expectations of these risks are infinite. In contrast, for concave functions of heavy-tailed risks with finite expectations, the stylized fact that diversification is preferable continues to hold. The framework of transformations of heavy-tailed risks includes many models with Pareto-type distributions that exhibit local or moderate deviations from power tails in the form of additional slowly varying or exponential factors. The class of distributions under study is therefore extended beyond the stable class.     

Second order freeness and fluctuations of random matrices: II. Unitary random matrices

We extend the relation between random matrices and free probability theory from the level of expectations to the level of fluctuations. We show how the concept of “second order freeness”, which was introduced in Part I, allows one to understand global fluctuations of Haar distributed unitary random matrices. In particular, independence between the unitary ensemble and another ensemble goes in the large N limit over into asymptotic second order freeness. Two important consequences of our general theory are: (i) we obtain a natural generalization of a theorem of Diaconis and Shahshahani to the case of several independent unitary matrices; (ii) we can show that global fluctuations in unitarily invariant multi-matrix models are not universal.     

Random fuzzy shock models and bivariate random fuzzy exponential distribution

In this paper, a random fuzzy shock model and a random fuzzy fatal shock model are proposed. Then bivariate random fuzzy exponential distribution is derived from the random fuzzy fatal shock model. Furthermore, some properties of the bivariate random fuzzy exponential distribution are proposed. Finally, an example is given to show the application of the bivariate random fuzzy exponential distribution.     

Optimizing an objective function under a bivariate probability model

The motivation of this paper is to obtain an analytical closed form of a quadratic objective function arising from a stochastic decision process with bivariate exponential probability distribution functions that may be dependent. This method is applicable when results need to be offered in an analytical closed form without double integrals. However, the study only applies to cases where the correlation coefficient between the two variables is positive or null. A stochastic, stationary objective function, involving a single decision variable in a quadratic form is studied. We use a primitive of a bivariate exponential distribution as first expressed by Downton [Downton, F., 1970. Bivariate exponential distributions in reliability theory. Journal of Royal Statistical Society B 32, 408–417] and revisited in Iliopoulos [Iliopoulos, George., 2003. Estimation of parametric functions in Downton’s bivariate exponential distribution. Journal of statistical planning and inference 117, 169–184]. With this primitive, optimization of objective functions in Operations Research, supply chain management or any other setting involving two random variables, or calculations which involve evaluating conditional expectations of two joint random variables are direct. We believe the results can be extended to other cases where exponential bivariates are encountered in economic objective function evaluations. Computation algorithms are offered which substantially reduce computation time when solving numerical examples.     

Vector-Valued Tail Value-at-Risk and Capital Allocation

Enterprise risk management, actuarial science or finance are practice areas in which risk measures are important to evaluate for heterogeneous classes of homogeneous risks. We present new measures: bivariate lower and upper orthant Tail Value-at-Risk. They are based on bivariate lower and upper orthant Value-at-Risk, introduced in Cossette et al. (Insurance: Math Econ 50(2):247–256, 2012). Many properties and applications are derived. Notably, they are shown to be positive homogeneous, invariant under translation and subadditive in distribution. Capital allocation criteria are suggested. Moreover, results on the sum of random pairs are presented, allowing to use a more accurate model for dependent classes of homogeneous risks.     

On the Use of Bivariate Mellin Transform in Bivariate Random Scaling and Some Applications

We discuss here the problem of bivariate random scaling. Both direct and inverse problems of bivariate random scaling are solved by two methods. While the first method is a distributional one, the second method is an indirect one associated with bivariate Mellin transform. Finally, we use bivariate random scaling for some statistical and simulational applications.     

Models for Stationary Max-Stable Random Fields

Models for stationary max-stable random fields are revisited and illustrated by two-dimensional simulations. We introduce a new class of models, which are based on stationary Gaussian random fields, and whose realizations are not necessarily semi-continuous functions. The bivariate marginal distributions of these random fields can be calculated, and they form a new class of bivariate extreme value distributions.     

Two parallel insurance lines with simultaneous arrivals and risks correlated with inter-arrival times

We investigate an insurance risk model that consists of two reserves which receive income at fixed rates. Claims are being requested at random epochs from each reserve and the interclaim times are generally distributed. The two reserves are coupled in the sense that at a claim arrival epoch, claims are being requested from both reserves and the amounts requested are correlated. In addition, the claim amounts are correlated with the time elapsed since the previous claim arrival.We focus on the probability that this bivariate reserve process survives indefinitely. The infinite-horizon survival problem is shown to be related to the problem of determining the equilibrium distribution of a random walk with vector-valued increments with ‘reflecting’ boundary. This reflected random walk is actually the waiting time process in a queueing system dual to the bivariate ruin process.Under assumptions on the arrival process and the claim amounts, and using Wiener–Hopf factorization with one parameter, we explicitly determine the Laplace–Stieltjes transform of the survival function, c.q., the two-dimensional equilibrium waiting time distribution.Finally, the bivariate transforms are evaluated for some examples, including for proportional reinsurance, and the bivariate ruin functions are numerically calculated using an efficient inversion scheme.     

On a property of bivariate distributions

Shanbag gave a characterization of the exponential and geometric distribution in terms of conditional expectations. Recently, Kotlarski generalized his method to obtain some properties of univariate probability distributions through conditional expectations. A property of bivariate distributions is given here generalizing Kotlarski's result in the univariate case.     

Second order freeness and fluctuations of random matrices: I. Gaussian and Wishart matrices and cyclic Fock spaces

We extend the relation between random matrices and free probability theory from the level of expectations to the level of fluctuations. We introduce the concept of “second order freeness” and interpret the global fluctuations of Gaussian and Wishart random matrices by a general limit theorem for second order freeness. By introducing cyclic Fock space, we also give an operator algebraic model for the fluctuations of our random matrices in terms of the usual creation, annihilation, and preservation operators. We show that orthogonal families of Gaussian and Wishart random matrices are asymptotically free of second order.     

Background risk,bivariate risk attitudes,and optimal prevention

This paper extends Eeckhoudt et al.’s (2012) results for precautionary effort to bivariate utility function framework. We establish an equivalence between the agent’s precautionary effort motive and the signs of successive cross-derivatives of the bivariate utility function. We show that the introduction (or deterioration) of an independent background risk induces more prevention to protect against wealth loss provided the individual exhibits correlation aversion of some given order. The conditions on the individual’s risk preferences are given to generate some specific prevention behaviors in the univariate framework with multiplicative risks. Our conclusion also indicates that an increase in the correlation between wealth risk and background risk leads to a reduction in optimal prevention.     

推广的相关序及其一些性质

本文考虑相关风险序，首先，把Dhaene和Goovaerts于1996年提出的相关序由二维随机向量推广到了多维随机向量的情况；然后，我们讨论了推广的相关序的一些性质；最后作为推论，我们还得到：由相关序可以推出指数序．     

两变量随机系数回归模型的最优设计

本文考虑两变量随机系数回归模型在单位正方形设计区域上基于A-,Ds-,I-和D-准则下的最优设计.证明了最优设计可在设计域的顶点处获得,并得到了几类最优设计的解析或数值结果.     

两变量随机截距模型的最优设计

本文考虑两变量随机截距模型在单位正方形设计域上的D-,G-,A-,Ds-和I-最优设计.证明了最优设计在设计区域的顶点处获得,得到了几类不依赖于随机截距项的最优设计模拟结果表明最优设计较随机设计可大幅提高参数估计的精确性.     

Sarmanov family of multivariate distributions for bivariate dynamic claim counts model

To predict future claims, it is well-known that the most recent claims are more predictive than older ones. However, classic panel data models for claim counts, such as the multivariate negative binomial distribution, do not put any time weight on past claims. More complex models can be used to consider this property, but often need numerical procedures to estimate parameters. When we want to add a dependence between different claim count types, the task would be even more difficult to handle. In this paper, we propose a bivariate dynamic model for claim counts, where past claims experience of a given claim type is used to better predict the other type of claims. This new bivariate dynamic distribution for claim counts is based on random effects that come from the Sarmanov family of multivariate distributions. To obtain a proper dynamic distribution based on this kind of bivariate priors, an approximation of the posterior distribution of the random effects is proposed. The resulting model can be seen as an extension of the dynamic heterogeneity model described in Bolancé et al. (2007). We apply this model to two samples of data from a major Canadian insurance company, where we show that the proposed model is one of the best models to adjust the data. We also show that the proposed model allows more flexibility in computing predictive premiums because closed-form expressions can be easily derived for the predictive distribution, the moments and the predictive moments.     

On sums of two counter-monotonic risks

In risk management, capital requirements are most often based on risk measurements of the aggregation of individual risks treated as random variables. The dependence structure between such random variables has a strong impact on the behavior of the aggregate loss. One finds an extensive literature on the study of the sum of comonotonic risks but less, in comparison, has been done regarding the sum of counter-monotonic risks. A crucial result for comonotonic risks is that the Value-at-risk and the Tail Value-at-risk of their sum correspond respectively to the sum of the Value-at-risk and Tail Value-at-risk of the individual risks. In this paper, our main objective is to derive such simple results for the sum of counter-monotonic risks. To do so, we examine separately different contexts in the class of bivariate strictly continuous distributions for which we obtain closed-form expressions for the Value-at-risk and Tail Value-at-risk of the sum of two counter-monotonic risks. The expressions for the subadditive Tail Value-at risk allow us to quantify the maximal diversification benefit. Also, our findings allow us to analyze the tail of the distribution of the sum of two identically subexponentially distributed counter-monotonic random variables.