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2018 | OriginalPaper | Chapter

4. Estimation

Authors : Marius Hofert, Ivan Kojadinovic, Martin Mächler, Jun Yan

Published in: Elements of Copula Modeling with R

Publisher: Springer International Publishing

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Abstract

This chapter addresses the estimation of copulas from a parametric, semi-parametric, and nonparametric perspective.

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Literature
go back to reference Berghaus, B., Bücher, A., & Volgushev, S. (2017). Weak convergence of the empirical copula process with respect to weighted metrics. Bernoulli, 23(1), 743–772.MathSciNetCrossRef Berghaus, B., Bücher, A., & Volgushev, S. (2017). Weak convergence of the empirical copula process with respect to weighted metrics. Bernoulli, 23(1), 743–772.MathSciNetCrossRef
go back to reference Capéraà, P., Fougères, A.-L., & Genest, C. (1997). A nonparametric estimation procedure for bivariate extreme value copulas. Biometrika, 84, 567–577.MathSciNetCrossRef Capéraà, P., Fougères, A.-L., & Genest, C. (1997). A nonparametric estimation procedure for bivariate extreme value copulas. Biometrika, 84, 567–577.MathSciNetCrossRef
go back to reference Carley, H., & Taylor, M. D. (2002). A new proof of Sklar’s theorem. In C. M. Cuadras, J. Fortiana, & J. A. Rodríguez-Lallena (Eds.), Distributions with given marginals and statistical modelling (pp. 29–34). Dordrecht: Kluwer Academic Publishers.CrossRef Carley, H., & Taylor, M. D. (2002). A new proof of Sklar’s theorem. In C. M. Cuadras, J. Fortiana, & J. A. Rodríguez-Lallena (Eds.), Distributions with given marginals and statistical modelling (pp. 29–34). Dordrecht: Kluwer Academic Publishers.CrossRef
go back to reference Charpentier, A., Fermanian, J.-D., & Scaillet, O. (2007). The estimation of copulas: Theory and practice. In J. Rank (Ed.), Copulas: From theory to application in finance (pp. 35–60). London: Risk Books. Charpentier, A., Fermanian, J.-D., & Scaillet, O. (2007). The estimation of copulas: Theory and practice. In J. Rank (Ed.), Copulas: From theory to application in finance (pp. 35–60). London: Risk Books.
go back to reference Chen, X., Fan, Y., & Tsyrennikov, V. (2006). Effcient estimation of semiparametric multivariate copula models. Journal of the American Statistical Association, 101, 1228–1240.MathSciNetCrossRef Chen, X., Fan, Y., & Tsyrennikov, V. (2006). Effcient estimation of semiparametric multivariate copula models. Journal of the American Statistical Association, 101, 1228–1240.MathSciNetCrossRef
go back to reference Deheuvels, P. (1979). La fonction de dépendance empirique et ses propriétés: un test non paramétrique d’indépendance. Académie Royale de Belgique Bulletin de la Classe des Sciences 5th Series, 65, 274–292.MATH Deheuvels, P. (1979). La fonction de dépendance empirique et ses propriétés: un test non paramétrique d’indépendance. Académie Royale de Belgique Bulletin de la Classe des Sciences 5th Series, 65, 274–292.MATH
go back to reference Deheuvels, P. (1981). A non parametric test for independence. Publications de l’Institut de Statistique de l’Université de Paris, 26, 29–50.MATH Deheuvels, P. (1981). A non parametric test for independence. Publications de l’Institut de Statistique de l’Université de Paris, 26, 29–50.MATH
go back to reference Deheuvels, P. (1991). On the limiting behavior of the Pickands estimator for bivariate extreme-value distributions. Statistics & Probability Letters, 12, 429–439.MathSciNetCrossRef Deheuvels, P. (1991). On the limiting behavior of the Pickands estimator for bivariate extreme-value distributions. Statistics & Probability Letters, 12, 429–439.MathSciNetCrossRef
go back to reference Demarta, S., & McNeil, A. J. (2005). The t copula and related copulas. International Statistical Review, 73(1), 111–129.CrossRef Demarta, S., & McNeil, A. J. (2005). The t copula and related copulas. International Statistical Review, 73(1), 111–129.CrossRef
go back to reference Embrechts, P., Lindskog, F., & McNeil, A. J. (2003). Modelling dependence with copulas and applications to risk management. In S. Rachev (Ed.), Handbook of heavy tailed distributions in finance (pp. 329–384). Amsterdam: Elsevier.CrossRef Embrechts, P., Lindskog, F., & McNeil, A. J. (2003). Modelling dependence with copulas and applications to risk management. In S. Rachev (Ed.), Handbook of heavy tailed distributions in finance (pp. 329–384). Amsterdam: Elsevier.CrossRef
go back to reference Fermanian, J.-D., Radulovic, D., & Wegkamp, M. (2004). Weak convergence of empirical copula processes. Bernoulli, 10(5), 847–860.MathSciNetCrossRef Fermanian, J.-D., Radulovic, D., & Wegkamp, M. (2004). Weak convergence of empirical copula processes. Bernoulli, 10(5), 847–860.MathSciNetCrossRef
go back to reference Fermanian, J.-D., & Scaillet, O. (2005). Some statistical pitfalls in copula modelling for financial applications. In E. Klein (Ed.), Capital formation, governance and banking (pp. 59–74). Hauppauge, NY: Nova Science. Fermanian, J.-D., & Scaillet, O. (2005). Some statistical pitfalls in copula modelling for financial applications. In E. Klein (Ed.), Capital formation, governance and banking (pp. 59–74). Hauppauge, NY: Nova Science.
go back to reference Gänssler, P., & Stute, W. (1987). Seminar on empirical processes, DMV Seminar 9. Basel: Birkhäuser. Gänssler, P., & Stute, W. (1987). Seminar on empirical processes, DMV Seminar 9. Basel: Birkhäuser.
go back to reference Genest, C., & Favre, A.-C. (2007). Everything you always wanted to know about copula modeling but were afraid to ask. Journal of Hydrological Engineering, 12, 347–368.CrossRef Genest, C., & Favre, A.-C. (2007). Everything you always wanted to know about copula modeling but were afraid to ask. Journal of Hydrological Engineering, 12, 347–368.CrossRef
go back to reference Genest, C., Ghoudi, K., & Rivest, L.-P. (1995). A semiparametric estimation procedure of dependence parameters in multivariate families of distributions. Biometrika, 82, 543–552.MathSciNetCrossRef Genest, C., Ghoudi, K., & Rivest, L.-P. (1995). A semiparametric estimation procedure of dependence parameters in multivariate families of distributions. Biometrika, 82, 543–552.MathSciNetCrossRef
go back to reference Genest, C., Kojadinovic, I., Nešlehová, J. G., & Yan, J. (2011). A goodness-of-fit test for bivariate extreme-value copulas. Bernoulli, 17(1), 253–275.MathSciNetCrossRef Genest, C., Kojadinovic, I., Nešlehová, J. G., & Yan, J. (2011). A goodness-of-fit test for bivariate extreme-value copulas. Bernoulli, 17(1), 253–275.MathSciNetCrossRef
go back to reference Genest, C., Masiello, E., & Tribouley, K. (2009). Estimating copula densities through wavelets. Insurance: Mathematics and Economics, 44, 170–181.MathSciNetMATH Genest, C., Masiello, E., & Tribouley, K. (2009). Estimating copula densities through wavelets. Insurance: Mathematics and Economics, 44, 170–181.MathSciNetMATH
go back to reference Genest, C., Nešlehová, J. G., & Rémillard, B. (2014). On the empirical multilinear copula process for count data. Bernoulli, 20, 1344–1371.MathSciNetCrossRef Genest, C., Nešlehová, J. G., & Rémillard, B. (2014). On the empirical multilinear copula process for count data. Bernoulli, 20, 1344–1371.MathSciNetCrossRef
go back to reference Genest, C., Nešlehová, J. G., & Rémillard, B. (2017). Asymptotic behavior of the empirical multilinear copula process under broad conditions. Journal of Multivariate Analysis, 20, 82–110.MathSciNetCrossRef Genest, C., Nešlehová, J. G., & Rémillard, B. (2017). Asymptotic behavior of the empirical multilinear copula process under broad conditions. Journal of Multivariate Analysis, 20, 82–110.MathSciNetCrossRef
go back to reference Genest, C., & Rivest, L.-P. (1993). Statistical inference procedures for bivariate Archimedean copulas. Journal of the American Statistical Association, 88, 1034–1043.MathSciNetCrossRef Genest, C., & Rivest, L.-P. (1993). Statistical inference procedures for bivariate Archimedean copulas. Journal of the American Statistical Association, 88, 1034–1043.MathSciNetCrossRef
go back to reference Genest, C., & Segers, J. (2009). Rank-based inference for bivariate extreme-value copulas. The Annals of Statistics, 37, 2990–3022.MathSciNetCrossRef Genest, C., & Segers, J. (2009). Rank-based inference for bivariate extreme-value copulas. The Annals of Statistics, 37, 2990–3022.MathSciNetCrossRef
go back to reference Genest, C., & Werker, B. J. M. (2002). Conditions for the asymptotic semiparametric efficiency of an omnibus estimator of dependence parameters in copula models. In C. M. Cuadras, J. Fortiana, & J. A. Rodríguez-Lallena (Eds.), Distributions with given marginals and statistical modelling (pp. 103–112). London: Kluwer Academic Publishers.CrossRef Genest, C., & Werker, B. J. M. (2002). Conditions for the asymptotic semiparametric efficiency of an omnibus estimator of dependence parameters in copula models. In C. M. Cuadras, J. Fortiana, & J. A. Rodríguez-Lallena (Eds.), Distributions with given marginals and statistical modelling (pp. 103–112). London: Kluwer Academic Publishers.CrossRef
go back to reference Gudendorf, G., & Segers, J. (2012). Nonparametric estimation of multivariate extreme-value copulas. Journal of Statistical Planning and Inference, 143, 3073–3085.MathSciNetCrossRef Gudendorf, G., & Segers, J. (2012). Nonparametric estimation of multivariate extreme-value copulas. Journal of Statistical Planning and Inference, 143, 3073–3085.MathSciNetCrossRef
go back to reference Higham, N. (2002). Computing the nearest correlation matrix – A problem from finance, IMA Journal of Numerical Analysis, 22, 329–343.MathSciNetCrossRef Higham, N. (2002). Computing the nearest correlation matrix – A problem from finance, IMA Journal of Numerical Analysis, 22, 329–343.MathSciNetCrossRef
go back to reference Janssen, P., Swanepoel, J., & Veraverbeke, N. (2012). Large sample behavior of the Bernstein copula estimator. Journal of Statistical Planning and Inference, 142, 1189–1197.MathSciNetCrossRef Janssen, P., Swanepoel, J., & Veraverbeke, N. (2012). Large sample behavior of the Bernstein copula estimator. Journal of Statistical Planning and Inference, 142, 1189–1197.MathSciNetCrossRef
go back to reference Joe, H. (1997). Multivariate models and dependence concepts. London: Chapman & Hall.CrossRef Joe, H. (1997). Multivariate models and dependence concepts. London: Chapman & Hall.CrossRef
go back to reference Joe, H. (2005). Asymptotic efficiency of the two-stage estimation method for copula-based models. Journal of Multivariate Analysis, 94, 401–419.MathSciNetCrossRef Joe, H. (2005). Asymptotic efficiency of the two-stage estimation method for copula-based models. Journal of Multivariate Analysis, 94, 401–419.MathSciNetCrossRef
go back to reference Joe, H., & Xu, J. J. (1996). The Estimation Method of Inference Functions for Margins for Multivariate Models. Technical Report, Department of Statistics, University of British Columbia. Joe, H., & Xu, J. J. (1996). The Estimation Method of Inference Functions for Margins for Multivariate Models. Technical Report, Department of Statistics, University of British Columbia.
go back to reference Kim, G., Silvapulle, M. J., & Silvapulle, P. (2007). Comparison of semiparametric and parametric methods for estimating copulas. Computational Statistics & Data Analysis, 51(6), 2836–2850.MathSciNetCrossRef Kim, G., Silvapulle, M. J., & Silvapulle, P. (2007). Comparison of semiparametric and parametric methods for estimating copulas. Computational Statistics & Data Analysis, 51(6), 2836–2850.MathSciNetCrossRef
go back to reference Klassen, C. A. J., & Wellner, J. A. (1997). Effcient estimation in the bivariate normal copula model: Normal marginals are least favourable. Bernoulli, 3, 55–77.MathSciNetCrossRef Klassen, C. A. J., & Wellner, J. A. (1997). Effcient estimation in the bivariate normal copula model: Normal marginals are least favourable. Bernoulli, 3, 55–77.MathSciNetCrossRef
go back to reference Kojadinovic, I., & Yan, J. (2010a). Comparison of three semiparametric methods for estimating dependence parameters in copula models. Insurance: Mathematics and Economics, 47, 52–63.MathSciNetMATH Kojadinovic, I., & Yan, J. (2010a). Comparison of three semiparametric methods for estimating dependence parameters in copula models. Insurance: Mathematics and Economics, 47, 52–63.MathSciNetMATH
go back to reference Kojadinovic, I., & Yan, J. (2010b). Nonparametric rank-based tests of bivariate extreme-value dependence. Journal of Multivariate Analysis, 101(9), 2234–2249.MathSciNetCrossRef Kojadinovic, I., & Yan, J. (2010b). Nonparametric rank-based tests of bivariate extreme-value dependence. Journal of Multivariate Analysis, 101(9), 2234–2249.MathSciNetCrossRef
go back to reference Lehmann, E. L., & Casella, G. (1998). Theory of point estimation. New York: Springer.MATH Lehmann, E. L., & Casella, G. (1998). Theory of point estimation. New York: Springer.MATH
go back to reference McNeil, A. J., Frey, R., & Embrechts, P. (2015). Quantitative risk management: Concepts, techniques and tools (2nd ed.). Princeton, NJ: Princeton University Press.MATH McNeil, A. J., Frey, R., & Embrechts, P. (2015). Quantitative risk management: Concepts, techniques and tools (2nd ed.). Princeton, NJ: Princeton University Press.MATH
go back to reference Oakes, D. (1982). A model for association in bivariate survival data. Journal of the Royal Statistical Society Series B, 44, 414–422.MathSciNetMATH Oakes, D. (1982). A model for association in bivariate survival data. Journal of the Royal Statistical Society Series B, 44, 414–422.MathSciNetMATH
go back to reference Omelka, M., Gijbels, I., & Veraverbeke, N. (2009). Improved kernel estimation of copulas: Weak convergence and goodness-of-fit testing. The Annals of Statistics, 37(5B), 3023–3058.MathSciNetCrossRef Omelka, M., Gijbels, I., & Veraverbeke, N. (2009). Improved kernel estimation of copulas: Weak convergence and goodness-of-fit testing. The Annals of Statistics, 37(5B), 3023–3058.MathSciNetCrossRef
go back to reference Pickands, J. (1981). Multivariate extreme value distributions. With a discussion. Proceedings of the 43rd session of the International Statistical Institute. Bulletin de l’Institut international de statistique, 49, 859–878, 894–902. Pickands, J. (1981). Multivariate extreme value distributions. With a discussion. Proceedings of the 43rd session of the International Statistical Institute. Bulletin de l’Institut international de statistique, 49, 859–878, 894–902.
go back to reference Rüschendorf, L. (1976). Asymptotic distributions of multivariate rank order statistics. The Annals of Statistics, 4, 912–923.MathSciNetCrossRef Rüschendorf, L. (1976). Asymptotic distributions of multivariate rank order statistics. The Annals of Statistics, 4, 912–923.MathSciNetCrossRef
go back to reference Ruymgaart, F. H. (1978). Asymptotic theory of ranks tests for independence. Amsterdam: Mathematisch Centrum. Ruymgaart, F. H. (1978). Asymptotic theory of ranks tests for independence. Amsterdam: Mathematisch Centrum.
go back to reference Sancetta, A., & Satchell, S. (2004). The Bernstein copula and its applications to modeling and approximations of multivariate distributions. Econometric Theory, 20, 535–562.MathSciNetCrossRef Sancetta, A., & Satchell, S. (2004). The Bernstein copula and its applications to modeling and approximations of multivariate distributions. Econometric Theory, 20, 535–562.MathSciNetCrossRef
go back to reference Segers, J. (2012). Asymptotics of empirical copula processes under nonrestrictive smoothness assumptions. Bernoulli, 18, 764–782.MathSciNetCrossRef Segers, J. (2012). Asymptotics of empirical copula processes under nonrestrictive smoothness assumptions. Bernoulli, 18, 764–782.MathSciNetCrossRef
go back to reference Segers, J., Sibuya, M., & Tsukahara, H. (2017). The empirical beta copula. Journal of Multivariate Analysis, 155, 35–51.MathSciNetCrossRef Segers, J., Sibuya, M., & Tsukahara, H. (2017). The empirical beta copula. Journal of Multivariate Analysis, 155, 35–51.MathSciNetCrossRef
go back to reference Shih, J. H., & Louis, T. A. (1995). Inferences on the association parameter in copula models for bivariate survival data. Biometrics, 51(4), 1384–1399.MathSciNetCrossRef Shih, J. H., & Louis, T. A. (1995). Inferences on the association parameter in copula models for bivariate survival data. Biometrics, 51(4), 1384–1399.MathSciNetCrossRef
go back to reference Stute, W. (1984). The oscillation behavior of empirical processes: The multivariate case. The Annals of Probability, 12(2), 361–379.MathSciNetCrossRef Stute, W. (1984). The oscillation behavior of empirical processes: The multivariate case. The Annals of Probability, 12(2), 361–379.MathSciNetCrossRef
go back to reference Tsukahara, H. (2005). Semiparametric estimation in copula models. The Canadian Journal of Statistics, 33(3), 357–375.MathSciNetCrossRef Tsukahara, H. (2005). Semiparametric estimation in copula models. The Canadian Journal of Statistics, 33(3), 357–375.MathSciNetCrossRef
go back to reference Van der Vaart, A. W., & Wellner, J. A. (2007). Empirical processes indexed by estimated functions. In E. A. Cator (Ed.), Asymptotics: Particles, processes and inverse problems (pp. 234–252). Lithuania: Institute of Mathematical Statistics.CrossRef Van der Vaart, A. W., & Wellner, J. A. (2007). Empirical processes indexed by estimated functions. In E. A. Cator (Ed.), Asymptotics: Particles, processes and inverse problems (pp. 234–252). Lithuania: Institute of Mathematical Statistics.CrossRef
Metadata
Title
Estimation
Authors
Marius Hofert
Ivan Kojadinovic
Martin Mächler
Jun Yan
Copyright Year
2018
DOI
https://doi.org/10.1007/978-3-319-89635-9_4