Estimation of extreme Component-wise Excess design realization: a hydrological application
The classic univariate risk measure in environmental sciences is the Return Period (RP). The RP is traditionally defined as “the average time elapsing between two successive realizations of a prescribed event”. The notion of design quantile related with RP is also of great importance. The design quantile represents the “value of the variable(s) characterizing the event associated with a given RP”. Since an individual risk may strongly be affected by the degree of dependence amongst all risks, the need for the provision of multivariate design quantiles has gained ground. In contrast to the univariate case, the design quantile definition in the multivariate setting presents certain difficulties. In particular, Salvadori, G., De Michele, C. and Durante F. define in the paper called “On the return period and design in a multivariate framework” (Hydrol Earth Syst Sci 15:3293–3305, 2011) the design realization as the vector that maximizes a weight function given that the risk vector belongs to a given critical layer of its joint multivariate distribution function. In this paper, we provide the explicit expression of the aforementioned multivariate risk measure in the Archimedean copula setting. Furthermore, this measure is estimated by using Extreme Value Theory techniques and the asymptotic normality of the proposed estimator is studied. The performance of our estimator is evaluated on simulated data. We conclude with an application on a real hydrological data-set.
KeywordsMultivariate design quantile Extreme Value Theory Return period Archimedean copula Hydrological application
The authors thank the associated editor and the referees whose comments helped to improve a previous version of this paper. Furthermore, the authors thank Gianfausto Salvadori and Fabrizio Durante for fruitful discussions. This work was partly supported by a grant from the Junta de Andalucía (Spain) for research group (FQM- 328) and by a pre-doctoral contract (Palacios Rodríguez, F.) from the “V Plan Propio de Investigación” of the University of Seville.
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