Max-stable processes are the natural extension of the classical extreme-value distributions to the functional setting, and they are increasingly widely used to estimate probabilities of complex extreme events. In this paper we broaden them from the usual situation in which dependence varies according to functions of Euclidean distance to situations in which extreme river discharges at two locations on a river network may be dependent because the locations are flow-connected or because of common meteorological events. In the former case dependence depends on river distance, and in the second it depends on the hydrological distance between the locations, either of which may be very different from their Euclidean distance. Inference for the model parameters is performed using a multivariate threshold likelihood, which is shown by simulation to work well. The ideas are illustrated with data from the upper Danube basin.Comment: Published at http://dx.doi.org/10.1214/15-AOAS863 in the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

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Abstract:We consider Wald's sequential probability ratio test for deciding whether a sequence of independent and identically distributed observations comes from a specified phase-type distribution or from an exponentially tilted alternative distribution. In this setting, we derive exact decision boundaries for given Type I and Type II errors by establishing a link with ruin theory. Information on the mean sample size of the test can be retrieved as well. The approach relies on the use of matrix-valued scale functions associated to a certain one-sided Markov additive process. By suitable transformations the results also apply to other types of distributions including some distributions with regularly varying tail.

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We consider Wald's sequential probability ratio test for deciding whether a sequence of independent and identically distributed observations comes from a specified phase-type distribution or from an exponentially tilted alternative distribution. In this setting, we derive exact decision boundaries for given Type I and Type II errors by establishing a link with ruin theory. Information on the mean sample size of the test can be retrieved as well. The approach relies on the use of matrix-valued scale functions associated to a certain one-sided Markov additive process. By suitable transformations the results also apply to other types of distributions including some distributions with regularly varying tail.

In this paper, we review and discuss some challenges in insuring flood risk in Europe on the national level, including high correlation of damages. Making use of recent advances in extreme value theory, we, furthermore, model flood risk with heavy-tailed distributions and their truncated counterparts and apply the discussed techniques to an inflation-and building-value-adjusted annual data set of flood losses in Europe. The analysis leads to Value-at-Risk estimates for individual countries and for Europe as a whole, allowing to quantify the diversification potential for flood risk in Europe. Finally, we identify optimal risk pooling possibilities in case a joint insurance strategy on the European level cannot be realized and quantify the resulting inefficiency in terms of additional necessary solvency capital. Thus, the results also contribute to the ongoing discussion on how public risk transfer mechanisms can supplement missing private insurance coverage. Keywords Flood risk Á Europe Á Risk pooling Á Risk transfer mechanism Á EVT techniques Á Value at Risk (VaR) Á Expected shortfall (ES) Á Joint risk pooling initiatives (JRPIs) Á Solvency II

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