, Volume 78, Issue 5, pp 497–526 | Cite as

Estimating covariate functions associated to multivariate risks: a level set approach

  • Elena Di Bernardino
  • Thomas Laloë
  • Rémi Servien


The aim of this paper is to study the behavior of a covariate function in a multivariate risks scenario. The first part of this paper deals with the problem of estimating the \(c\)-upper level sets \({L(c)= \{F(x) \ge c \}}\), with \(c \in (0,1)\), of an unknown distribution function \(F\) on \(\mathbb {R}^d_+\). A plug-in approach is followed. We state consistency results with respect to the volume of the symmetric difference. In the second part, we obtain the \(L_p\)-consistency, with a convergence rate, for the regression function estimate on these level sets \(L(c)\). We also consider a new multivariate risk measure: the Covariate-Conditional-Tail-Expectation. We provide a consistent estimator for this measure with a convergence rate. We propose a consistent estimate when the regression cannot be estimated on the whole data set. Then, we investigate the effects of scaling data on our consistency results. All these results are proven in a non-compact setting. A complete simulation study is detailed and a comparison with parametric and semi-parametric approaches is provided. Finally, a real environmental application of our risk measure is provided.


Multidimensional distribution function Plug-in estimation  Regression function 



The authors thank the two anonymous reviewers and an associated editor for their useful comments and suggestions. This work has been partially supported by the French research national agency (ANR) under the references ANR-08BLAN-0314-01 and ANR 2011 BS01 010 01 projet Calibration. The authors thank Yannick Baraud, Roland Diel, Christine Tuleau-Malo and Patricia Reynaud-Bouret for fruitful discussions.


  1. Artzner P, Delbaen F, Eber JM, Heath D (1999) Coherent measures of risk. Math Finance 9(3):203–228. doi: 10.1111/1467-9965.00068 MATHMathSciNetCrossRefGoogle Scholar
  2. Baíllo A, Cuesta-Albertos J, Cuevas A (2001) Convergence rates in nonparametric estimation of level sets. Stat Probab Lett 53:27–35MATHCrossRefGoogle Scholar
  3. Belzunce F, Castaño A, Olvera-Cervantes A, Suárez-Llorens A (2007) Quantile curves and dependence structure for bivariate distributions. Comput Stat Data Anal 51(10):5112–5129. doi: 10.1016/j.csda.2006.08.017 MATHCrossRefGoogle Scholar
  4. Bentahar I (2006) Tail conditional expectation for vector-valued risks. Sfb 649 discussion papers, Humboldt University, Collaborative Research Center 649.
  5. Billingsley P (1995) Probability and measure, 3rd edn. Wiley Series in Probability and Mathematical Statistics. Wiley, a Wiley-Interscience Publication, New YorkGoogle Scholar
  6. Cai J, Li H (2005) Conditional tail expectations for multivariate phase-type distributions. J Appl Probab 42(3):810–825MATHMathSciNetCrossRefGoogle Scholar
  7. Cavalier L (1997) Nonparametric estimation of regression level sets. Stat A J Theor Appl Stat 29(2):131–160. doi: 10.1080/02331889708802579 MATHMathSciNetGoogle Scholar
  8. Chebana F, Ouarda T (2011) Multivariate quantiles in hydrological frequency analysis. Environmetrics 22(1):63–78. doi: 10.1002/env.1027 MathSciNetCrossRefGoogle Scholar
  9. Cousin A, Di Bernardino E (2013) On multivariate extensions of Value-at-Risk. J Mulitivar Anal 119:32–46MATHCrossRefGoogle Scholar
  10. Cuevas A, González-Manteiga W, Rodríguez-Casal A (2006) Plug-in estimation of general level sets. Aust N Z J Stat 48(1):7–19. doi: 10.1111/j.1467-842X.2006.00421.x MATHMathSciNetCrossRefGoogle Scholar
  11. Daouia A, Gardes L, Girard S, Lekina A (2010) Kernel estimators of extreme level curves. Test 20:311–333MathSciNetCrossRefGoogle Scholar
  12. Dedu S, Ciumara R (2010) Restricted optimal retention in stop-loss reinsurance under VaR and CTE risk measures. Proc Rom Acad Ser A Math Phys Tech Sci Inf Sci 11(3):213–217MathSciNetGoogle Scholar
  13. Dekens B, Meerdink L, Meijer G, Sirks E, van Vliet R (2011) Vung Tau - Go Cong Dam Vietnam, Preliminary Design Study, Final report: Project Group Flood Defence, Department of Hydraulic Engineering, Delft University of Technology.,d.ZGU
  14. Denuit M, Dhaene J, Goovaerts M, Kaas R (2005) Actuarial theory for dependent risks. Wiley, New YorkCrossRefGoogle Scholar
  15. Di Bernardino E, Laloë T, Maume-Deschamps V, Prieur C (2013) Plug-in estimation of level sets in a non-compact setting with applications in multivariable risk theory. ESAIM: Probab Stat 17:236–256MATHMathSciNetCrossRefGoogle Scholar
  16. Draisma G, Drees H, Ferreira A, de Haan L (2004) Bivariate tail estimation: dependence in asymptotic independence. Bernoulli 10(2):251–280. doi: 10.3150/bj/1082380219 MATHMathSciNetCrossRefGoogle Scholar
  17. Ekeland I, Galichon A, Henry M (2012) Comonotonic measures of multivariate risks. Technical reportGoogle Scholar
  18. Embrechts P, Puccetti G (2006) Bounds for functions of multivariate risks. J Multivar Anal 97(2):526–547. doi: 10.1016/j.jmva.2005.04.001 MATHMathSciNetCrossRefGoogle Scholar
  19. Fan J, Yao Q (2003) Nonlinear time series. Springer series in statistics. Springer, New YorkGoogle Scholar
  20. de Haan L, Huang X (1995) Large quantile estimation in a multivariate setting. J Multivar Anal 53(2):247–263. doi: 10.1006/jmva.1995.1035 MATHCrossRefGoogle Scholar
  21. Hawkes J, Gouldby BP, Tawn JA, Owen MW (2002) The joint probability ofwaves andwater levels in coastal engineering design. J Hydraul Res 40(3):241–251CrossRefGoogle Scholar
  22. Imlahi L, Ezzerg M, Chakak A (1999) Estimación de la curva mediana de una cópula \({C}(x_1, \ldots, x_n)\). Rev R Acad Cien Exact Fis Nat 93(2):241–250MATHMathSciNetGoogle Scholar
  23. Jouini E, Meddeb M, Touzi N (2004) Vector-valued coherent risk measures. Finance Stoch 8(4):531–552. doi: 10.1007/s00780-004-0127-6 MATHMathSciNetCrossRefGoogle Scholar
  24. Kohler M, Krzyzak A, Walk H (2009) Optimal global rates of convergence for nonparametric regression with unbounded data. J Stat Plan Inference 139(4):1286–1296. doi: 10.1016/j.jspi.2008.07.012 MATHMathSciNetCrossRefGoogle Scholar
  25. Laloë T, Servien R (2013) Nonparametric estimation of regression level sets using kernel plug-in estimator. JKSS 42(3):301–311. doi: 10.1016/j.jkss.2012.10.001 MATHGoogle Scholar
  26. Mason M, Polonik W (2009) Asymptotic normality of plug-in level set estimates. Ann Appl Probab 19:1108–1142MATHMathSciNetCrossRefGoogle Scholar
  27. Nappo G, Spizzichino F (2009) Kendall distributions and level sets in bivariate exchangeable survival models. Inf Sci 179:2878–2890. doi: 10.1016/j.ins.2009.02.007.
  28. Nelsen RB (1999) An introduction to copulas, Lecture Notes in Statistics, vol 139. Springer, New YorkCrossRefGoogle Scholar
  29. Rigollet P, Vert R (2009) Optimal rates for plug-in estimators of density level sets. Bernoulli 15(4):1154–1178. doi: 10.3150/09-BEJ184 MATHMathSciNetCrossRefGoogle Scholar
  30. Salvadori G, De Michele C, Durante F (2011) On the return period and design in a multivariate framework. Hydrol Earth Syst Sci 15(11):3293–3305. doi: 10.5194/hess-15-3293-2011 CrossRefGoogle Scholar
  31. Tibiletti L (1993) On a new notion of multidimensional quantile. Metron Int J Stat 51(3–4):77–83MATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Elena Di Bernardino
    • 1
  • Thomas Laloë
    • 2
  • Rémi Servien
    • 3
  1. 1.Conservatoire National des Arts et Métiers, IMATH, EA4629ParisFrance
  2. 2.CNRS, LJAD, UMR 7351Université de Nice Sophia-AntipolisNiceFrance
  3. 3.INRA, UMR1331 Toxalim, Research Centre in Food ToxicologyUniversite de ToulouseToulouseFrance

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