@TECHREPORT{Bentahar, title = {Tail {C}onditional {E}xpectation for vector-valued Risks}, author = {Bentahar, Imen}, year = {2006}, institution = {Humboldt University, Collaborative Research Center 649}, type = {SFB 649 Discussion Papers}, number = {SFB649DP2006-029}, abstract = {In his paper we introduce a quantile-based risk measure for multivariate financial positions "the vector-valued Tail-conditional-expectation (TCE)". We adopt the framework proposed by Jouini, Meddeb, and Touzi [9] to deal with multi-assets portfolios when one accounts for frictions in the financial market. In this framework, the space of risks formed by essentially bounded random vectors, is endowed with some partial vector preorder >= accounting for market frictions. In a first step we provide a definition for quantiles of vector-valued risks which is compatible with the preorder >=. The TCE is then introduced as a natural extension of the "classical" real-valued tail-conditional-expectation. Our main result states that for continuous distributions TCE is equal to a coherent vector-valued risk measure. We also provide a numerical algorithm for computing vector-valued quantiles and TCE.}, keywords = {Risk measures; vector-valued risk measures; coherent risk-measures; quantiles; tail-conditional-expectation}, url = {http://econpapers.repec.org/RePEc:hum:wpaper:sfb649dp2006-029} } @article{MAO, title = "Second-order properties of the {H}aezendonck-{G}oovaerts risk measure for extreme risks", journal = "Insurance: Mathematics and Economics", volume = "51", number = "2", pages = "333 - 343", year = "2012", issn = "0167-6687", doi = "https://doi.org/10.1016/j.insmatheco.2012.06.003", url = "http://www.sciencedirect.com/science/article/pii/S0167668712000704", author = "Tiantian Mao and Taizhong Hu", keywords = "(Extended) regular variation, Extreme value theory, First-order expansion, Max-domain attraction, Second-order expansion, Second-order regular variation, Young function" } @article{Weissman, ISSN = {01621459}, URL = {http://www.jstor.org/stable/2286285}, abstract = {We consider an estimation problem when only the k largest observations of a sample of size n are available. It is assumed that the underlying distribution function F belongs to the domain of attraction of a known extreme-value distribution and that k remains fixed as n → ∞. We present estimators for the location and scale parameters and for p-quantiles of F, where p is of the form 1 - c/n (c fixed). These estimators are either asymptotically maximum likelihood or minimum variance.}, author = {Ishay Weissman}, journal = {Journal of the American Statistical Association}, number = {364}, pages = {812-815}, publisher = {[American Statistical Association, Taylor & Francis, Ltd.]}, title = {Estimation of Parameters and Larger Quantiles Based on the k Largest Observations}, volume = {73}, year = {1978} } @article{draisma2004, author = "Draisma, Gerrit and Drees, Holger and Ferreira, Ana and De Haan, Laurens", doi = "10.3150/bj/1082380219", fjournal = "Bernoulli", journal = "Bernoulli", month = "04", number = "2", pages = "251--280", publisher = "Bernoulli Society for Mathematical Statistics and Probability", title = "Bivariate tail estimation: dependence in asymptotic independence", url = "https://doi.org/10.3150/bj/1082380219", volume = "10", year = "2004" } @article{Salvadori3, author = {G. Salvadori and F. Durante and E. Perrone}, title = {Semi-parametric approximation of {K}endall's distribution function and multivariate {R}eturn {P}eriods}, journal = {Journal de la Soci\'{e}t\'{e} Fran\c{c}aise de Statistique}, volume={154}, number = {1}, year = {2013}, PAGES = {151--173} } @book{Salvadori2, AUTHOR = {Salvadori, G. and De Michele, C. and Kottegoda, N.T. and Rosso, R.}, TITLE = {Extremes in Nature: An Approach Using {C}opulas.}, PUBLISHER = {Springer-Verlag: Berlin}, YEAR = {2007}, } @Article{Vandenberghe, AUTHOR = {Vandenberghe, S., and van den Berg, M. J., and Gr\"{a}ler, B., and Petroselli, A., and Grimaldi, S., and De Baets, B., and Verhoest, N. E. C.}, TITLE = {Joint return periods in hydrology: a critical and practical review focusing on synthetic design hydrograph estimation}, JOURNAL = { Hydrology and Earth System Sciences}, VOLUME = {9}, YEAR = {2012}, PAGES = {6781--6828} } @Article{Graler, AUTHOR = {Gr\"aler, B. and van den Berg, M. J. and Vandenberghe, S. and Petroselli, A. and Grimaldi, S. and De Baets, B. and Verhoest, N. E. C.}, TITLE = {Multivariate return periods in hydrology: a critical and practical review focusing on synthetic design hydrograph estimation}, JOURNAL = {Hydrology and Earth System Sciences}, VOLUME = {17}, YEAR = {2013}, NUMBER = {4}, PAGES = {1281--1296}, URL = {http://www.hydrol-earth-syst-sci.net/17/1281/2013/}, DOI = {10.5194/hess-17-1281-2013} } @article {Einmahl2006, AUTHOR = {Einmahl, John H. J. and de Haan, Laurens and Li, Deyuan}, TITLE = {Weighted approximations of tail copula processes with application to testing the bivariate extreme value condition}, JOURNAL = {Ann. Statist.}, FJOURNAL = {The Annals of Statistics}, VOLUME = {34}, YEAR = {2006}, NUMBER = {4}, PAGES = {1987--2014}, ISSN = {0090-5364}, CODEN = {ASTSC7}, MRCLASS = {62G32 (60F17 60G70 62G10 62G30)}, MRNUMBER = {2283724 (2008e:62092)}, MRREVIEWER = {Anne-Laure Foug{\`e}res}, DOI = {10.1214/009053606000000434}, URL = {http://dx.doi.org/10.1214/009053606000000434}, } @article{van2007empirical, title={Empirical processes indexed by estimated functions}, author={van der Vaart, Aad W and Wellner, Jon A}, journal={Lecture Notes-Monograph Series}, pages={234--252}, year={2007}, publisher={JSTOR} } @article{Qu, title = "Copula density estimation by total variation penalized likelihood with linear equality constraints ", journal = "Computational Statistics \& Data Analysis ", volume = "56", number = "2", pages = "384 - 398", year = "2012", note = "", issn = "0167-9473", doi = "10.1016/j.csda.2011.07.016", url = "http://www.sciencedirect.com/science/article/pii/S0167947311002830", author = "Leming Qu and Wotao Yin", keywords = "Copula density estimation", keywords = "Total variation", keywords = "Maximum penalized likelihood estimation", keywords = "Augmented Lagrangian method " } @article{Lambert, title = "Archimedean copula estimation using {B}ayesian splines smoothing techniques ", journal = "Computational Statistics \& Data Analysis ", volume = "51", number = "12", pages = "6307 - 6320", year = "2007", note = "", issn = "0167-9473", doi = "10.1016/j.csda.2007.01.018", url = "http://www.sciencedirect.com/science/article/pii/S0167947307000217", author = "Philippe Lambert", keywords = "Archimedean copula", keywords = "Bayesian P-splines", keywords = "Markov chains Monte Carlo", keywords = "Monotonicity and convexity constraints " } @book {ReissThomas, AUTHOR = {Reiss, R.-D. and Thomas, M.}, TITLE = {Statistical analysis of extreme values with applications to insurance, finance, hydrology and other fields}, EDITION = {Third}, NOTE = {With 1 CD-ROM (Windows)}, PUBLISHER = {Birkh\"auser Verlag}, ADDRESS = {Basel}, YEAR = {2007}, } @book {NelsenLibro, AUTHOR = {Nelsen, Roger B.}, TITLE = {An introduction to copulas}, SERIES = {Lecture Notes in Statistics}, VOLUME = {139}, PUBLISHER = {Springer-Verlag}, ADDRESS = {New York}, YEAR = {1999}, PAGES = {xii+216}, ISBN = {0-387-98623-5}, MRCLASS = {60E05 (60E15 62H05)}, MRNUMBER = {1653203 (99i:60028)}, MRREVIEWER = {Moshe Shaked}, } @TechReport{Galichon, author={Henry, Marc and Galichon, Alfred and Ekeland, Ivar}, title={Comonotonic Measures of Multivariate Risks}, year=2012, month= , institution={Université Paris-Dauphine}, type={Open Access publications from Université Paris-Dauphine}, url={http://ideas.repec.org/p/ner/dauphi/urnhdl123456789-2278.html}, number={urn:hdl:123456789/2278}, abstract={We propose a multivariate extension of a well-known characterization by S. Kusuoka of regular and coherent risk measures as maximal correlation functionals. This involves an extension of the notion of comonotonicity to random vectors through generalized quantile functions. Moreover, we propose to replace the current law invari- ance, subadditivity and comonotonicity axioms by an equivalent property we call strong coherence and that we argue has more natural economic interpretation. Finally, we refor- mulate the computation of regular and coherent risk measures as an optimal transportation problem, for which we provide an algorithm and implementation.}, keywords={Comonotonicity; Maximal Correlation; Optimal Transportation; Regular Risk Measures; Coherent Risk Me} } @book {Feller, AUTHOR = {Feller, William}, TITLE = {An introduction to probability theory and its applications. {V}ol. {II}}, PUBLISHER = {John Wiley \& Sons Inc.}, ADDRESS = {New York}, YEAR = {1966}, PAGES = {xviii+636}, MRCLASS = {60.00}, MRNUMBER = {0210154 (35 \#1048)}, MRREVIEWER = {S. 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We adopt the framework proposed by Jouini, Meddeb, and Touzi [9] to deal with multi-assets portfolios when one accounts for frictions in the financial market. In this framework, the space of risks formed by essentially bounded random vectors, is endowed with some partial vector preorder >= accounting for market frictions. In a first step we provide a definition for quantiles of vector-valued risks which is compatible with the preorder >=. The TCE is then introduced as a natural extension of the "classical" real-valued tail-conditional-expectation. Our main result states that for continuous distributions TCE is equal to a coherent vector-valued risk measure. We also provide a numerical algorithm for computing vector-valued quantiles and TCE.}, keywords = {Risk measures; vector-valued risk measures; coherent risk-measures; quantiles; tail-conditional-expectation}, url = {http://econpapers.repec.org/RePEc:hum:wpaper:sfb649dp2006-029} } @article{Gaenssler, AUTHOR = {Gaenssler, P., and Stute, W.}, TITLE = {Empirical processes: {A} survey of results for independent and identically distributed random variables}, JOURNAL = {Ann. Probab.}, VOLUME = {7}, YEAR = {1979}, PAGES = {193--243}, } @article {Belzunce, AUTHOR = {Belzunce, F. and Casta{\~n}o, A. and Olvera-Cervantes, A. and Su{\'a}rez-Llorens, A.}, TITLE = {Quantile curves and dependence structure for bivariate distributions}, JOURNAL = {Comput. Statist. 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Arcones}, DOI = {10.1016/B978-044450083-0/50012-5}, URL = {http://dx.doi.org/10.1016/B978-044450083-0/50012-5}, } @book {deHaanlibro, AUTHOR = {de Haan, Laurens and Ferreira, Ana}, TITLE = {Extreme value theory}, SERIES = {Springer Series in Operations Research and Financial Engineering}, NOTE = {An introduction}, PUBLISHER = {Springer}, ADDRESS = {New York}, YEAR = {2006}, } @unpublished{article2, hal_id = {hal-00834000}, url = {http://hal.archives-ouvertes.fr/hal-00834000}, title = {{On certain transformations of {A}rchimedean copulas : {A}pplication to the non-parametric estimation of their generators}}, author = {Di Bernardino, Elena and Rulli{\`e}re, Didier}, keywords = {Distortions; Archimedean copula; auto-nested copula; non-parametric estimation; tail dependence}, language = {Anglais}, affiliation = {Centre d'Etude et De Recherche en Informatique du Cnam - CEDRIC , Laboratoire de Sciences Actuarielle et Financi{\`e}re - SAF}, year = {2013}, month = Jun, pdf = {http://hal.archives-ouvertes.fr/hal-00834000/PDF/Article\_DiBernardino\_Rulliere\_2013\_Hal\_v1.pdf}, } @article {Acharya, AUTHOR = {Acharya, V.V., and Pedersen, L.H., and Philippon, T., and Richardson, M.}, TITLE = {Measuring systemic risk}, JOURNAL = {Preprint}, YEAR = {2012}, } @article {Hurlimann, AUTHOR = {Werner H\"{u}rlimann}, TITLE = {On some properties of two vector valued {V}a{R} and {CTE} multivariate risk measures for {A}rchimedean copulas}, JOURNAL = {Astin Bulletin}, VOLUME = {44}, NUMBER = {3}, PAGES = {613--633}, YEAR = {2014}, } @article {ENV:ENV2385, author = {Di Bernardino, E. and Palacios-Rodr\'{\i}guez, F.}, title = {Estimation of extreme quantiles conditioning on multivariate critical layers}, journal = {Environmetrics}, volume = {27}, number = {3}, issn = {1099-095X}, url = {http://dx.doi.org/10.1002/env.2385}, doi = {10.1002/env.2385}, pages = {158--168}, keywords = {multivariate risk measures, return levels, critical layers, extreme quantile}, year = {2016}, } @article{Salvadori2014guidelines, Author = {Salvadori, G. and Tomasicchio, G. R. and D'Alessandro, F.}, Journal = {Coastal Engineering }, Title = {Practical guidelines for multivariate analysis and design in coastal and off-shore engineering.}, Year = {2014}, Volume={88}, Pages={1-14}} @article{Salvadori2011, Author = {Salvadori, G. and De Michele, C. and Durante, F.}, Journal = {Hydrology and Earth System Sciences}, Title = {On the return period and design in a multivariate framework.}, Year = {2011}, Volume={15}, Pages={3293-3305} } @book{Singh2007, Author = {Singh, V. and Jain, S. and Tyagi, A.}, Publisher = {ASCE Press, Reston, Virginia.}, Title = {Risk and Reliability Analysis.}, Year = {2007}} @Article{Bernardino2017, author="Di Bernardino, E. and Palacios-Rodr{\'i}guez, F.", title="Estimation of extreme Component-wise Excess design realization: a hydrological application", journal="Stochastic Environmental Research and Risk Assessment", year="2017", month="Feb", day="09", abstract="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.", issn="1436-3259", doi="10.1007/s00477-017-1387-y", url="https://doi.org/10.1007/s00477-017-1387-y" } @article {Brownlees, AUTHOR = {Brownlees, C., and Engle, R.,}, TITLE = {Volatility, correlation and tails for systemic risk measurement}, JOURNAL = {Preprint}, YEAR = {2012}, } C. Brownlees and R. Engle. Volatility, correlation and tails for systemic risk measurement. 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Un enfoque geometrico}, author = {Rodr\'{\i}guez-Casal, Alberto}, school = {University of Santiago de Compostela}, year = {2003} } @PhDThesis{DiBernardinothesis, title = {Mod\'{e}lisation de la d\'{e}pendance et mesures de risque multidimensionnelles}, author = {Di Bernardino, Elena}, school = {University of Lyon 1, http://isfaserveur.univ-lyon1.fr/\~elena.dibernardino/TESI}, year = {2011} } @inproceedings{Venter, AUTHOR = {Venter, G.G}, TITLE = {Tails of copulas}, booktitle = {Proceedings ASTIN Washington}, YEAR = {2001}, PAGES = {68--113}, } @article{Durante2014, title = "Copulas, diagonals, and tail dependence ", journal = "Fuzzy Sets and Systems ", year = "2014", issn = "0165-0114", doi = "http://dx.doi.org/10.1016/j.fss.2014.03.014", url = "http://www.sciencedirect.com/science/article/pii/S0165011414001353", author = "Fabrizio Durante and Juan Fern\'{a}ndez-S\'{a}nchez and Roberta Pappad\`{a}", keywords = "Risk aggregation", keywords = "Copulas", keywords = "Diagonal section", keywords = "Risk management", keywords = "Tail dependence ", abstract = "Abstract We present some known and novel aspects about bivariate copulas with prescribed diagonal section by highlighting their use in the description of the tail dependence. Moreover, we present the tail concentration function (which depends on the diagonal section of a copula) as a tool to give a description of tail dependence at finite scale. The tail concentration function is hence used to introduce a graphical tool that can help to distinguish different families of copulas in the copula test space. Moreover, it serves as a basis to determine the grouping structure of different financial time series by taking into account their pairwise tail behavior. " } @book{gorge2013insurance, title={Insurance Risk Management and Reinsurance}, author={Gorge, G.}, url={http://books.google.fr/books?id=2lEWBQAAQBAJ}, year={2013}, publisher={Gorge}, } @article{hill1975, author = "Hill, Bruce M.", doi = "10.1214/aos/1176343247", fjournal = "The Annals of Statistics", journal = "Ann. 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