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Title: On the random number of multivariate risks Authors:  Simone Padoan - Bocconi University (Italy) [presenting]
Enkelejd Hashorva - University of Lausanne (Switzerland)
Stefano Rizzelli - EPFL (Switzerland)
Abstract: Modeling of joint extreme-values is of interest in many real applications. Consider for simplicity a bivariate random vector $(X_1,X_2)$, with joint distribution function $F$, representing two quantities of interest and let $(X_{1,1},X_{1,2})$, $\ldots$, $(X_{n,1},X_{n,2})$ be iid copies of it. Let $N$ be a non-negative random variable representing the total number of events that have occurred during the period under investigation. Recent contributions investigate the extremal dependence underlying the distribution function, denoted by $H$, of the compound random variables $\left(\max_{1 \le i \le N} X_{i,1}, \max_{1 \le i \le N} X_{i,2}\right)$. Under appropriate conditions on $N$, it has been shown that the extremal properties of the distribution $H$ are the same as those of the distribution $F$. An open question we are interested in is the reverse problem. Specifically, we address the following question: what are the extremal properties of $F$ given that those of $H$ are known? We answer this question by exploiting the multivariate extreme-value theory. We provide the conditions on $N$ under which extremal properties of $H$ are different from those of $F$. Starting from an estimator of the extremal dependence of $H$ we derive an estimator for the extremal dependence of $F$. For the latter estimator we derive the asymptotic properties and for finite samples its performance is illustrated via a simulation study.