Hostname: page-component-8448b6f56d-dnltx Total loading time: 0 Render date: 2024-04-18T04:30:07.619Z Has data issue: false hasContentIssue false
  • English
  • Français

Plug-in estimation of level sets in a non-compact setting with applications in multivariate risk theory

Published online by Cambridge University Press:  08 February 2013

Elena Di Bernardino ,
Thomas Laloë ,
Véronique Maume-Deschamps  and
Clémentine Prieur
Elena Di Bernardino
Affiliation:
Universitéde Lyon, Université Lyon 1, ISFA, Laboratoire SAF, 50 Avenue Tony Garnier, 69366 Lyon, France. elena.di-bernardino@univ-lyon1.fr; veronique.maume@univ-lyon1.fr
Thomas Laloë
Affiliation:
Université de Nice Sophia-Antipolis, Laboratoire J-A Dieudonné, Parc Valrose, 06108 Nice Cedex 02, France; thomas.laloe@unice.fr
Véronique Maume-Deschamps
Affiliation:
Universitéde Lyon, Université Lyon 1, ISFA, Laboratoire SAF, 50 Avenue Tony Garnier, 69366 Lyon, France. elena.di-bernardino@univ-lyon1.fr; veronique.maume@univ-lyon1.fr
Clémentine Prieur
Affiliation:
Université Joseph Fourier, Tour IRMA, MOISE-LJK B.P. 53 38041 Grenoble, France; clementine.prieur@imag.fr
Get access

Abstract

This paper deals with the problem of estimating the level sets L(c) =  {F(x) ≥ c}, with c ∈ (0,1), of an unknown distribution function F on ℝ+2. A plug-in approach is followed. That is, given a consistent estimator Fn of F, we estimate L(c) by Ln(c) =  {Fn(x) ≥ c}. In our setting, non-compactness property is a priori required for the level sets to estimate. We state consistency results with respect to the Hausdorff distance and the volume of the symmetric difference. Our results are motivated by applications in multivariate risk theory. In particular we propose a new bivariate version of the conditional tail expectation by conditioning the two-dimensional random vector to be in the level set L(c). We also present simulated and real examples which illustrate our theoretical results.

Keywords

Level setsdistribution functionplug-in estimationHausdorff distanceconditional tail expectation
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 17 , 2013 , pp. 236 - 256
Copyright
© EDP Sciences, SMAI, 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Références

Artzner, P., Delbaen, F., Eber, J.-M. and Heath, D., Coherent measures of risk. Math. Finance 9 (1999) 203–228. Google Scholar
Baíllo, A., Total error in a plug-in estimator of level sets. Statist. Probab. Lett. 65 (2003) 411–417. Google Scholar
Baíllo, A., Cuesta-Albertos, J.A. and Cuevas, A., Convergence rates in nonparametric estimation of level sets. Statist. Probab. Lett. 53 (2001) 27–35. Google Scholar
Belzunce, F., Castaño, A., Olvera-Cervantes, A. and Suárez-Llorens, A., Quantile curves and dependence structure for bivariate distributions. Comput. Stat. Data Anal. 51 (2007) 5112–5129. Google Scholar
Biau, G., Cadre, B. and Pelletier, B., A graph-based estimator of the number of clusters. ESAIM : PS 11 (2007) 272–280. Google Scholar
P. Billingsley, Probability and measure. Wiley Series in Probability and Mathematical Statistics, 3th edition, John Wiley & Sons Inc., A Wiley-Interscience Publication, New York (1995).
Cadre, B., Kernel estimation of density level sets. J. Multivar. Anal. 97 (2006) 999–1023. Google Scholar
Cai, J. and Li, H., Conditional tail expectations for multivariate phase-type distributions. J. Appl. Probab. 42 (2005) 810–825. Google Scholar
Cavalier, L., Nonparametric estimation of regression level sets. Statistics (Berl. DDR) 29 (1997) 131–160. Google Scholar
Chaubey, Y.P. and Sen, P.K., Smooth estimation of multivariate survival and density functions. J. Statist. Plann. Inference 103 (2002) 361–376; C. R. Rao 80th birthday felicitation volume, Part I. Google Scholar
Cuevas, A. and Fraiman, R., A plug-in approach to support estimation. Ann. Stat. 25 (1997) 2300–2312. Google Scholar
Cuevas, A. and Rodríguez–Casal, A., On boundary estimation. Adv. Appl. Probab. 36 (2004) 340–354. Google Scholar
Cuevas, A., González-Manteiga, W. and Rodríguez–Casal, A., Plug-in estimation of general level sets. Australian & New Zealand J. Statist. 48 (2006) 7–19. Google Scholar
de Haan, L. and Huang, X., Large quantile estimation in a multivariate setting. J. Multivar. Anal. 53 (1995) 247–263. Google Scholar
Dedu, S. and Ciumara, R., Restricted optimal retention in stop-loss reinsurance under VaR and CTE risk measures. Proc. of Rom. Acad. Ser. A 11 (2010) 213–217. Google Scholar
M. Denuit, J. Dhaene, M. Goovaerts and R. Kaas, Actuarial Theory for Dependent Risks. Wiley, (2005).
Embrechts, P. and Puccetti, G., Bounds for functions of multivariate risks. J. Multivar. Anal. 97 (2006) 526–547. Google Scholar
Fernández-Ponce, J.M. and Suárez-Lloréns, A., Central regions for bivariate distributions. Austrian J. Stat. 31 (2002) 141–156. Google Scholar
Frees, E.W. and Valdez, E.A., Understanding relationships using copulas. North Amer. Actuar. J. 2 (1998) 1–25. Google Scholar
Hartigan, J.A., Estimation of a convex density contour in two dimensions. J. Amer. Statist. Assoc. 82 (1987) 267–270. Google Scholar
Koltchinskii, V.I., M-estimation, convexity and quantiles. Ann. Statist. 25 (1997) 435–477. Google Scholar
T. Laloë, Sur Quelques Problèmes d’Apprentissage Supervisé et Non Supervisé. Ph.D. thesis, University Montpellier II (2009).
Massé, J.-C. and Theodorescu, R., Halfplane trimming for bivariate distributions. J. Multivar. Anal. 48 (1994) 188–202. Google Scholar
Nappo, G. and Spizzichino, F., Kendall distributions and level sets in bivariate exchangeable survival models. Inform. Sci. 179 (2009) 2878–2890. Google Scholar
Polonik, W., Measuring mass concentrations and estimating density contour clusters – an excess mass approach. Ann. Stat. 23 (1995) 855–881. Google Scholar
Polonik, W., Minimum volume sets and generalized quantile processes. Stoch. Proc. Appl. 69 (1997) 1–24. Google Scholar
Rigollet, P. and Vert, R., Optimal rates for plug-in estimators of density level sets. Bernoulli. 15 (2009) 1154–1178. Google Scholar
A. Rodríguez–Casal. Estimacíon de conjuntos y sus fronteras. Un enfoque geometrico. Ph.D. thesis, University of Santiago de Compostela (2003).
Rossi, C., Sulle curve di livello di una superficie di ripartizione in due variabili; on level curves of two dimensional distribution function. Giornale dell’Istituto Italiano degli Attuari 36 (1973) 87–108. Google Scholar
Rossi, C., Proprietà geometriche delle superficie di ripartizione. Rend. Mat. (6) 9 (1976) 725–736 (1977). Google Scholar
Serfling, R., Quantile functions for multivariate analysis : approaches and applications. Stat. Neerlandica 56 (2002) 214–232 Special issue : Frontier research in theoretical statistics (2000) (Eindhoven). Google Scholar
Tibiletti, L., On a new notion of multidimensional quantile. Metron 51 (1993) 77–83. Google Scholar
Tsybakov, A.B., On nonparametric estimation of density level sets. Ann. Stat. 25 (1997) 948–969. Google Scholar