Modelling the extremal dependence of bivariate variables is important in a wide variety of practical applications, including environmental planning, catastrophe modelling and hydrology. The majority of these approaches are based on the framework of bivariate regular variation, and a wide range of literature is available for estimating the dependence structure in this setting. However, this framework is only applicable to variables exhibiting asymptotic dependence, even though asymptotic independence is often observed in practice. In this paper, we consider the so-called 'angular dependence function'; this quantity summarises the extremal dependence structure for asymptotically independent variables. Until recently, only pointwise estimators of the angular dependence function have been available. We introduce a range of global estimators and compare them to another recently introduced technique for global estimation through a systematic simulation study, and a case study on river flow data from the north of England, UK.
In the multivariate setting, defining extremal risk measures is important in many contexts, such as finance, environmental planning and structural engineering. In this paper, we review the literature on extremal bivariate return curves, a risk measure that is the natural bivariate extension to a return level, and propose new estimation methods based on multivariate extreme value models that can account for both asymptotic dependence and asymptotic independence. We identify gaps in the existing literature and propose novel tools for testing and validating return curves and comparing estimates from a range of multivariate models. These tools are then used to compare a selection of models through simulation and case studies. We conclude with a discussion and list some of the challenges.
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