Extremes of a Random Number of Multivariate Risks
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Risk analysis in the area of insurance, financial and risk management is concerned with the study of the joint probability that multiple extreme events take place simultaneously. Extreme Value Theory provides tools for estimating such a type of probability. When aggregate data such as maxima of a random number of observations (risks), few results are available, and this is especially true in high dimensions. To fill this gap, we derive the asymptotic distribution of normalized maxima of multivariate risks, under appropriate conditions on the random number of observations. Of the latter we derive the extremal dependence which is for heavy-tailed scenarios stronger than the dependence among the usual multivariate maxima described in the classical theory. We establish the connection between the two dependence structures through random scaling and Pickands dependence functions. By means of the so-called inverse problem, we construct a semiparametric estimator for the extremal dependence of the unobservable data, starting from an estimator of the extremal dependence obtained with the aggregated data. We develop the asymptotic theory of the estimator and further explore by a simulation study its finite-sample performance.
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