Comparing generalized Pareto models fitted to extreme observations: an application to the largest temperatures in Spain
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In this paper, a subsampling-based testing procedure for the comparison of the exceedance distributions of stationary time series is introduced. The proposed testing procedure has a number of advantages including the fact that the assumption of stationary can be relaxed for some specific forms of non-stationary and also that the two time series are not required to be independently-generated. For this purpose, a test based on the Kolmogorov–Smirnov and the L 1-Wasserstein distances between generalized Pareto distributions is introduced and studied in some detail. The performance of the testing procedure is illustrated through a simulation study and with an empirical application to a set of data concerning daily maximum temperature in the 17 autonomous communities of Spain for the period 1990–2004. The autonomous communities were clustered according to the similarities of the fitted generalized Pareto models and then mapped. The cluster analysis reveals a clear distinction between the four northeast communities on the shores of the Bay of Biscay (which are the regions exhibiting milder temperatures) and the remaining regions. A second cluster corresponds to the southern Mediterranean area and the central region which corresponds to the communities with highest temperatures.
KeywordsExtreme value theory Generalized Pareto distribution Peaks over threshold Subsampling Stationarity and non-stationary time series
The authors would like to express their gratitude to the editor and the two referees. They offered extremely valuable perspectives on our work and effective suggestions for improvements. The second author would like to thank the support of the Department of Statistics of the University Carlos III, Madrid (Spain), where she stayed during her last 4 months of Sabbatical Leave. This stay was financially supported by the grant SFRH/BSAB/ 1138/2011, from Fundação para a Ciência e Tecnologia-FCT, Portugal. The research was also funded by FCT projects PEst-OE/MAT/UI0006/2011 and PTDC/MAT/118335/2010. The first author also acknowledges the support of CICYT (Spain) Grants SEJ2007-64500, ECO2011-25706 and ECO2012-38442. The third author was supported by FEDER funds through COMPETE–Operational Programme Factors of Competitiveness (“Programa Operacional Factores de Competitividade”) and by Portuguese funds through the Center for Research and Development in Mathematics and Applications (University of Aveiro) and the Portuguese Foundation for Science and Technology (“FCT–Fundação para a Ciência e a Tecnologia”), within project PEst-C/MAT/UI4106/2011 with COMPETE number FCOMP-01-0124-FEDER-022690.
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