Copulas: a Review and Recent Developments (2007)

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0 2 4 6 8 10 120 1 2 3 4 5 60 2 4 6 8 10Daily returns of Korea & Singapore0 1 2 3 4 5 6Standardized residuals of Korea & SingaporeFigure 1: Excesses (left) and standardized residuals excesses (right)Because of space limitations, it was impossible even to comment the di±cultieswith copulas associated to discrete multivariate distributions, the important aspects incopula theory related to stochastic ordering, e.g. MÄuller and Stoyan (2001), graphicaltechniques, e.g. Genest and Boies (2003), classical and new measures of dependence,e.g. Nelsen (1999) and Genest and Plante (2003), etc. A part of these interestingtopics are discussed in Anjos et al. (2004). We would like to mention ¯nally theimportance of survival copulas in life modelling. An excellent overview is given byGeorges et al. (2001), see also Nelsen (2005b).6 AknowledgementsApartofthepresentedmaterialinSection3was,infact,aplanned¯rststepsoftheresearch project related with a visit of the ¯rst author in CIMAT, Guanajuato, Mexicoin the beginning of 2003. As a result, in Kolev et al. (2006) is given an alternativede¯nition of the probability generating function of arbitrary binary vector, wherecomonotonic and countermonotonic discrete copulas play a key role.Special thanks to the organizers of the 8th Symposium of Probability and StochasticProcesses provided in Puebla, Mexico, June 20-25, 2004, where the authors havebeen invited to present their recent results.The authors are thankful to Professor Roger Nelsen for the continuous attentionon our research. We are greatful to the annonimous referee whose critical remarksreduced the errors and improved this overview.The ¯rst and the third authors are partially supported by FAPESP, Grant 03/10105-2 and PROBRAL (CAPES/DAAD), Grant 171-04. The second author thanks for a¯nancial support from CNPq and FAPERJ Grant E-26/170.725/2004.36
ReferencesAlsina,C.,Nelsen,R.B.,Schweizer,B.(1993). On the characterization of a class ofbinary operations on distribution functions. Statistics & Probability Letters 17,85-89.Ane, T., Kharoubi, C. (2003). Dependence structure and risk measure. Journal ofBusiness 76, 411-438.Ang, A., Chen, J., Xing, Y. (2002). Downside correlation and expected stock returns.Working paper No. 01-25, Columbia Business School. Available athttp://ssrn.com/abstract=282986Anjos,U.,Ferreira,F.,Kolev,N.,Mendes,B.(2004). Dependence Modelling viaCopulas. Short course in 16th SINAPE, Caxambu-MG, Brazil. (in Portuguese,English version under preparation). University Press USP: S~ao Paulo.Anjos, U., Kolev, N., Tanaka, N. (2005). Copula associated to order statistics.Brazilian Journal of Probability and Statistics 19, 111-123.Anjos, U., Kolev, N. (2005a). An application of Kendall distribution. Forthcomingin Journal for Economy and Management L(1), 95-101.Anjos, U., Kolev, N. (2005b). Representation of bivariate copulas via local measureof dependence. Technical Report RT-MAE 2005-03, University of S~ao Paulo.Artzner, P., Delbean, F., Eber, J.M., Heath, D. (1999). Coherent measures of risk.Mathematical Finance 9, 203-228.Averous, J., Genest, C., Kochar, S. (2005). On the dependence structure of orderstatistics. Journal of Multivariate Analysis 94, 159-171.Baillie, R.T., Bollerslev, T., Mikkelsen, H.O. (1996). Fractionally integrated generalizedautoregressive conditional heteroskedasticity. Journal of Econometrics74, 3-30.Barakat, H. (2001). The asymptotic distribution theory of bivariate order statistics.Ann. Inst. Stat. Math. 53, 487-497.Balakrishnan, N., Cohen, A.C. (1991). Order Statistics and Inference. AcademicPress: San Diego.Bjerve, S., Doksum, K. (1993). Correlation curves: measures of association as functionsof covariate values. Annals of Statistics 21, 890-902.Bollerslev, T., Mikkelsen, H.O. (1999). Long-term equity anticipation securities andstock market volatility dynamics. Journal of Econometrics 92, 75-99.37
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