Multivariate Compound Distributions Research Articles

Composite likelihood estimation method for hierarchical Archimedean copulas defined with multivariate compound distributions

We consider the family of hierarchical Archimedean copulas obtained from multivariate exponential mixture distribution through compounding, as introduced by Cossette et al. (2017). We investigate ways of determining the structure of these copulas and estimating their parameters. An agglomerative clustering technique based on the matrix of Spearman’s rhos, combined with a bootstrap procedure, is used to identify the tree structure. Parameters are estimated through a top-down composite likelihood. The validity of the approach is illustrated through two simulation studies in which the procedure is explained step by step. The composite likelihood method is also compared to the full likelihood method in a simple case where the latter is computable.

On the evaluation of some multivariate compound distributions with Sarmanov’s counting distribution

In this paper, we consider Sarmanov’s multivariate discrete distribution as counting distribution in two multivariate compound models: the first model assumes different types of independent claim sizes (corresponding to, e.g., different types of insurance policies), while in the second model, we introduce some dependency between the claims (motivated by the events that can simultaneously affect several types of policies). Since Sarmanov’s distribution can join different types of marginals, we also assume that these marginals belong to Panjer’s class of distributions and discuss the evaluation of the resulting compound distribution based on recursions. Alternatively, the evaluation of the same distribution using the Fast Fourier Transform method is also presented, with the purpose to significantly reduce the computing time, especially in the dependency case. Both methods are numerically illustrated and compared from the point of view of speed and accuracy.

ON THE EVALUATION OF MULTIVARIATE COMPOUND DISTRIBUTIONS WITH CONTINUOUS SEVERITY DISTRIBUTIONS AND SARMANOV'S COUNTING DISTRIBUTION

AbstractIn this paper, we present closed-type formulas for some multivariate compound distributions with multivariate Sarmanov counting distribution and independent Erlang distributed claim sizes. Further on, we also consider a type-II Pareto dependency between the claim sizes of a certain type. The resulting densities rely on the special hypergeometric function, which has the advantage of being implemented in the usual software. We numerically illustrate the applicability and efficiency of such formulas by evaluating a bivariate cumulative distribution function, which is also compared with the similar function obtained by the classical recursion-discretization approach.

Hierarchical Archimedean copulas through multivariate compound distributions

In this paper, we propose a new hierarchical Archimedean copula construction based on multivariate compound distributions. This new imbrication technique is derived via the construction of a multivariate exponential mixture distribution through compounding. The absence of nesting and marginal conditions, contrarily to the nested Archimedean copulas approach, leads to major advantages, such as a flexible range of possible combinations in the choice of distributions, the existence of explicit formulas for the distribution of the sum, and computational ease in high dimensions. A balance between flexibility and parsimony is targeted. After presenting the construction technique, properties of the proposed copulas are investigated and illustrative examples are given. A detailed comparison with other construction methodologies of hierarchical Archimedean copulas is provided. Risk aggregation under this newly proposed dependence structure is also examined.

On the Evaluation of Some Multivariate Compound Distributions with Sarmanov's Counting Distribution

In this paper, we consider Sarmanov's multivariate discrete distribution as counting distribution in two multivariate compound models: the First model assumes different types of independent claim sizes (corresponding to, e.g., different types of insurance policies), while in the second model, we introduce some dependency between the claims (motivated by the events that can simultaneously affect several types of policies). Since Sarmanov's distribution can join different types of marginals, we also assume that these marginals belong to Panjer's class of distributions and discuss the evaluation of the resulting compound distribution based on recursions. Alternatively, the evaluation of the same distribution using the Fast Fourier Transform method is also presented, with the purpose to significantly reduce the computing time, especially in the dependency case. Both methods are numerically illustrated and compared from the point of view of speed and accuracy.

TVaR-based capital allocation for multivariate compound distributions with positive continuous claim amounts

In this paper, we consider a portfolio of n dependent risks X1,…,Xn and we study the stochastic behavior of the aggregate claim amount S=X1+⋯+Xn. Our objective is to determine the amount of economic capital needed for the whole portfolio and to compute the amount of capital to be allocated to each risk X1,…,Xn. To do so, we use a top–down approach. For (X1,…,Xn), we consider risk models based on multivariate compound distributions defined with a multivariate counting distribution. We use the TVaR to evaluate the total capital requirement of the portfolio based on the distribution of S, and we use the TVaR-based capital allocation method to quantify the contribution of each risk. To simplify the presentation, the claim amounts are assumed to be continuously distributed. For multivariate compound distributions with continuous claim amounts, we provide general formulas for the cumulative distribution function of S, for the TVaR of S and the contribution to each risk. We obtain closed-form expressions for those quantities for multivariate compound distributions with gamma and mixed Erlang claim amounts. Finally, we treat in detail the multivariate compound Poisson distribution case. Numerical examples are provided in order to examine the impact of the dependence relation on the TVaR of S, the contribution to each risk of the portfolio, and the benefit of the aggregation of several risks.

On the Multivariate Compound Distributions

We present two methods of constructing multivariate compound distributions and investigate the corresponding infinitely divisible and compound Poisson distributions. We then show that the multivariate compound Poisson distributions can be derived as the limiting distributions of the sums of independent random vectors.

On the Likelihood Ratio Test for Compound Distributions for Homogeneity of Mixing Proportions

We consider the problem of comparing two multivariate compound distributions each with component densities f 1, (x) and f 2(x) and with mixing proportions p x and (1 – px) for the first compound distribution and py and (1 – py ) for the second. The likelihood ratio test of the hypothesis H 0: px = py is presented by using the EM algorithm to derive the maximum likelihood estimates. This test is applied to some actual data under the assumption that the underlying component densities are normal. The result is contrasted with results obtained using some existing univariate methods of testing H0.