Tail Behaviour of Weighted Sums of Order Statistics of Dependent Risks

Let $X_{1},\ldots ,X_{n}$ be $n$ real-valued dependent random variables. With motivation from Mitra and Resnick (2009), we derive the tail asymptotic expansion for the weighted sum of order statistics $X_{1:n}\leq \cdots \leq X_{n:n}$ of $X_{1},\ldots ,X_{n}$ under the general case in which the distribution function of $X_{n:n}$ is long-tailed or rapidly varying and $X_{1},\ldots ,X_{n}$ may not be comparable in terms of their tail probability. We also present two examples and an application of our results in risk theory

Aggregation of randomly weighted large risks

Asymptotic tail probabilities for linear combinations of randomly weighted order statistics are approximated under various assumptions. One key assumption is the asymptotic independence for all risks. Therefore, it is not surprising that the maxima represents the most influential factor when one investigates the tail behaviour of our considered risk aggregation, which, for example, can be found in the reinsurance market. This extreme behaviour confirms the ‘one big jump’ property that has been vastly discussed in the existing literature in various forms whenever asymptotic independence is present. An illustration of our results together with a specific application are explored under the assumption that the underlying risks follow the multivariate log-normal distribution

Tail asymptotics of randomly weighted large risks

Tail asymptotic probabilities for linear combinations of randomly weighted order statistics are approximated under various assumptions. One key assumption is the asymptotic independence for all risks, and thus, it is not surprising that the maxima represents the most influential factor when one investigates the tail behaviour of our considered risk aggregation, which for example, can be found in the reinsurance market. This extreme behaviour confirms the “one big jump” property that has been vastly discussed in the existing literature in various forms whenever the asymptotic independence is present. An illustration of our results together with a specific application are explored under the assumption that the underlying risks follow the multivariate Log-normal distribution. Keywords and phrases: Davis-Resnick tail property; Extreme value distribution; Max-domain of attraction; Mitra-Resnick model; Risk aggregatio

Efficient simulation of tail probabilities for sums of log-elliptical risks

In the framework of dependent risks it is a crucial task for risk management purposes to quantify the probability that the aggregated risk exceeds some large value u. Motivated by Asmussen et al. (2011) [1] in this paper we introduce a modified Asmussen-Kroese estimator for simulation of the rare event that the aggregated risk exceeds u. We show that in the framework of log-Gaussian risks our novel estimator has the best possible performance i.e., it has asymptotically vanishing relative error. For the more general class of log-elliptical risks with marginal distributions in the Gumbel max-domain of attraction we propose a modified Rojas-Nandayapa estimator of the rare events of interest, which for specific importance sampling densities has a good logarithmic performance. Our numerical results presented in this paper demonstrate the excellent performance of our novel Asmussen-Kroese algorithm