Skip to content
BY-NC-ND 4.0 license Open Access Published by De Gruyter Open Access July 1, 2017

On Truncation Invariant Copulas and their Estimation

  • Piotr Jaworski EMAIL logo
From the journal Dependence Modeling


The paper deals with the family of irreducible left truncation invariant bivariate copulas, which admit a nontrivial lower tail dependence function. Such copulas, similarly as the Archimedean ones, are characterized by a functional parameter, a generator being an increasing convex function.We provide a nonparametric, piece-wise linear estimator of such generators.


[1] Charpentier, A. and A. Juri (2006). Limiting dependence structures for tail events, with applications to credit derivatives. J. Appl. Probab. 43(2), 563-586.10.1239/jap/1152413742Search in Google Scholar

[2] Charpentier, A. and J. Segers (2009). Tails of multivariate Archimedean copulas. J. Multivariate Anal. 100(7), 1521-1537.10.1016/j.jmva.2008.12.015Search in Google Scholar

[3] Cherubini, U., E. Luciano, and W. Vecchiato (2004). Copula Methods in Finance. John Wiley & Sons, Chichester.10.1002/9781118673331Search in Google Scholar

[4] Cormier, E., C. Genest and J.G. Nešlehová (2014). Using B-splines for nonparametric inference on bivariate extreme-value copulas. Extremes 17(4), 633-659.10.1007/s10687-014-0199-4Search in Google Scholar

[5] Di Lascio, F.M.L., F. Durante and P. Jaworski (2016). Truncation invariant copulas and a testing procedure. J. Stat. Comput. Simul. 86(12), 2362-2378.10.1080/00949655.2015.1110820Search in Google Scholar

[6] Di Lascio, F.M.L., F. Durante and P. Jaworski (2017). A test for truncation invariant dependence. In Soft Methods for Data Science, pp. 173-180, Springer.10.1007/978-3-319-42972-4_22Search in Google Scholar

[7] Durante, F. and P. Jaworski (2012). Invariant dependence structure under univariate truncation. Statistics 46(2), 263-277.10.1080/02331888.2010.512977Search in Google Scholar

[8] Durante, F. and C. Sempi (2016). Principles of Copula Theory. CRC Press, Boca Raton FL.10.1201/b18674Search in Google Scholar

[9] Durante, F., P. Jaworski, and R. Mesiar (2011). Invariant dependence structures and Archimedean copulas. Stat. Probabil. Lett. 81(12), 1995-2003.10.1016/j.spl.2011.08.018Search in Google Scholar

[10] Embrechts, P. (2009). Copulas: A personal view. J. Risk Insur. 76(3), 639-650.10.1111/j.1539-6975.2009.01310.xSearch in Google Scholar

[11] Jaworski, P. (2003). Asymptotyka dwuwymiarowych kopuli. Matematyka Stosowana 4, 78-89.Search in Google Scholar

[12] Jaworski, P. (2004). On uniform tail expansions of bivariate copulas. Appl. Math. 31(4), 397-415.10.4064/am31-4-2Search in Google Scholar

[13] Jaworski, P. (2006). On uniform tail expansions of multivariate copulas and wide convergence of measures. Appl. Math. 33(2), 159-184.10.4064/am33-2-3Search in Google Scholar

[14] Jaworski, P. (2010). Tail behaviour of copulas. In Copula Theory and its Applications, pp. 161-186. Springer, Heidelberg.10.1007/978-3-642-12465-5_8Search in Google Scholar

[15] Jaworski, P. (2013). Invariant dependence structure under univariate truncation: the high-dimensional case. Statistics 47(5), 1064-1074.10.1080/02331888.2012.664143Search in Google Scholar

[16] Jaworski, P. (2013). The limiting properties of copulas under univariate conditioning. In Copulae inMathematical and Quantitative Finance, pp. 129-163. Springer, Heidelberg.10.1007/978-3-642-35407-6_7Search in Google Scholar

[17] Jaworski, P., F. Durante, W. Härdle, and T. Rychlik, editors (2010). Copula Theory and its Applications. Springer, Heidelberg.10.1007/978-3-642-12465-5Search in Google Scholar

[18] Jaworski, P., F. Durante, andW. Härdle, editors (2013). Copulae inMathematical and Quantitative Finance. Springer, Heidelberg.10.1007/978-3-642-35407-6Search in Google Scholar

[19] Joe, H. (1997). Multivariate Models and Dependence Concepts. Chapman & Hall, London.10.1201/9780367803896Search in Google Scholar

[20] Joe, H. (2014). Dependence Modeling with Copulas. Chapman & Hall/CRC, Boca Raton FL.10.1201/b17116Search in Google Scholar

[21] Joe, H., H. Li, and A.K. Nikoloulopoulos (2010). Tail dependence functions and vine copulas. J. Multivariate Anal. 101 (1), 252-270.10.1016/j.jmva.2009.08.002Search in Google Scholar

[22] Juri, A. and M.V. Wüthrich (2002). Copula convergence theorems for tail events. Insurance Math. Econom. 30(3), 405-420.10.1016/S0167-6687(02)00121-XSearch in Google Scholar

[23] Li, H. and P. Wu (2013). Extremal dependence of copulas: a tail density approach. J. Multivariate Anal. 114(1), 99-111.10.1016/j.jmva.2012.07.005Search in Google Scholar

[24] Łojasiewicz, S. (1988). An Introduction to the Theory of Real Functions. Third edition. John Wiley & Sons, Chichester.Search in Google Scholar

[25] Nelsen, R.B. (2006). An Introduction to Copulas. Second edition. Springer, New York.Search in Google Scholar

Received: 2016-12-31
Accepted: 2017-6-12
Published Online: 2017-7-1
Published in Print: 2017-1-26

© 2017

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Downloaded on 31.5.2023 from
Scroll to top button