Second order risk aggregation with the Bernstein copula

Coqueret, Guillaume

doi:10.1016/j.insmatheco.2014.07.002

Cited by 3 publications

(6 citation statements)

References 20 publications

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“…Although it can be easily obtained for independent risks, this assumption is in most cases too restrictive; thus, being crucial to specify more general models that allow for dependence between different risks. In the recent statistical and actuarial literature, several results about risk aggregation under dependence have been obtained, which deploy different copula structures (see, e.g., Arbenz et al (2012), Coqueret (2014) and Gijbels and Herrmann (2014)). Cossette et al (2013) consider risk aggregation and capital allocation problems for a portfolio of dependent risks, modelling the multivariate distribution with the Farlie-Gumbel-Morgenstern (FGM) copula and mixed Erlang distribution marginals.…”

Section: Introductionmentioning

confidence: 99%

Aggregation of Dependent Risks in Mixtures of Exponential Distributions and Extensions

Sarabia

Gómez–Déniz²,

Prieto³

et al. 2018

ASTIN Bull.

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The distribution of the sum of dependent risks is a crucial aspect in actuarial sciences, risk management and in many branches of applied probability. In this paper, we obtain analytic expressions for the probability density function (pdf) and the cumulative distribution function (cdf) of aggregated risks, modeled according to a mixture of exponential distributions. We first review the properties of the multivariate mixture of exponential distributions, to then obtain the analytical formulation for the pdf and the cdf for the aggregated distribution. We study in detail some specific families with Pareto (Sarabia et al, 2016), Gamma, Weibull and inverse Gaussian mixture of exponentials (Whitmore and Lee, 1991) claims. We also discuss briefly the computation of risk measures, formulas for the ruin probability (Albrecher et al., 2011) and the collective risk model. An extension of the basic model based on mixtures of gamma distributions is proposed, which is one of the suggested directions for future research. .es (V. Jordá). The modelLet Θ be a positive random variable with cdf F Θ (·) and Laplace-Stieltjes transform (hereinafter referred to as Laplace transform) L Θ (·), that is L Θ (s) = E[exp(−sΘ)] = ∞ 0 e −sz dF Θ (z). A distribution with support on (0, ∞) is identified by its Laplace transform. We consider the classical compound

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Section: Introductionmentioning

confidence: 99%

Aggregation of Dependent Risks in Mixtures of Exponential Distributions and Extensions

Sarabia

Gómez–Déniz²,

Prieto³

et al. 2018

ASTIN Bull.

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“…In model (8), each marginal component is distributed according to a second kind beta distribution. Indeed, X i is a second kind beta distribution with shape parameters p i and q 0 , scale parameter λ i and location parameter τ i , i = 1, 2, .…”

Section: Multivariate Extensionmentioning

confidence: 99%

“…. , X n be a random sample of d−dimensional vectors from (8). The maximum likelihood estimators of the parameters are given by,…”

Section: Parameter Estimationmentioning

confidence: 99%

“…, Theorem 3. Let X be a random vector defined by the stochastic representation (8), and consider the distribution of the aggregated risks…”

Section: Distribution Of the Aggregated Risksmentioning

confidence: 99%

“…Dependence of losses can be introduced via copulas. [6][7][8][9][10] Popular multivariate distributions for losses are: the multivariate Pareto distribution, 11 multivariate mixtures of exponential distributions 12 and the multivariate second kind beta distribution. 13 The widespread use of copula functions can only be understood because it is a simple way to overcome the connection between the marginal distributions and the multivariate distribution, while keeping mathematical expressions tractable.…”

Section: Introductionmentioning

confidence: 99%

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Aggregation of Dependent Risks with Heavy-Tail Distributions

Guillén

Sarabia

Prieto

et al. 2019

Int. J. Unc. Fuzz. Knowl. Based Syst.

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Straightforward methods to evaluate risks arising from several sources are specially difficult when risk components are dependent and, even more if that dependence is strong in the tails. We give an explicit analytical expression for the probability distribution of the sum of non-negative losses that are tail-dependent. Our model allows dependence in the extremes of the marginal beta distributions. The proposed model is flexible in the choice of the parameters in the marginal distribution. The estimation using the method of moments is possible and the calculation of risk measures is easily done with a Monte Carlo approach. An illustration on data for insurance losses is presented.

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Copula Representations for the Sum of Dependent Risks: Models and Comparisons

Navarro¹,

Sarabia²

2020

Prob. Eng. Inf. Sci.

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The study of the distributions of sums of dependent risks is a key topic in actuarial sciences, risk management, reliability and in many branches of applied and theoretical probability. However, there are few results where the distribution of the sum of dependent random variables is available in a closed form. In this paper, we obtain several analytical expressions for the distribution of the aggregated risks under dependence in terms of copulas. We provide several representations based on the underlying copula and the marginal distribution functions under general hypotheses and in any dimension. Then, we study stochastic comparisons between sums of dependent risks. Finally, we illustrate our theoretical results by studying some specific models obtained from Clayton, Ali-Mikhail-Haq and Farlie-Gumbel-Morgenstern copulas. Extensions to more general copulas are also included. Bounds and the limiting behavior of the hazard rate function for the aggregated distribution of some copulas are studied as well.

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Second order risk aggregation with the Bernstein copula

Cited by 3 publications

References 20 publications

Aggregation of Dependent Risks in Mixtures of Exponential Distributions and Extensions

Aggregation of Dependent Risks in Mixtures of Exponential Distributions and Extensions

Aggregation of Dependent Risks with Heavy-Tail Distributions

Copula Representations for the Sum of Dependent Risks: Models and Comparisons

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