Plug-in estimation of level sets in a non-compact setting with applications in multivariate risk theory
ESAIM: Probability and Statistics, Tome 17 (2013), pp. 236-256.

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.

DOI : 10.1051/ps/2011161
Classification : 62G05, 62G20, 60E05, 91B30
Mots clés : level sets, distribution function, plug-in estimation, Hausdorff distance, conditional tail expectation 
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     author = {Di Bernardino, Elena and Lalo\"e, Thomas and Maume-Deschamps, V\'eronique and Prieur, Cl\'ementine},
     title = {Plug-in estimation of level sets in a non-compact setting with applications in multivariate risk theory},
     journal = {ESAIM: Probability and Statistics},
     pages = {236--256},
     publisher = {EDP-Sciences},
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     mrnumber = {3021318},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/ps/2011161/}
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Di Bernardino, Elena; Laloë, Thomas; Maume-Deschamps, Véronique; Prieur, Clémentine. Plug-in estimation of level sets in a non-compact setting with applications in multivariate risk theory. ESAIM: Probability and Statistics, Tome 17 (2013), pp. 236-256. doi : 10.1051/ps/2011161. http://www.numdam.org/articles/10.1051/ps/2011161/

[1] P. Artzner, F. Delbaen, J.-M. Eber and D. Heath, Coherent measures of risk. Math. Finance 9 (1999) 203-228. | MR | Zbl

[2] A. Baíllo, Total error in a plug-in estimator of level sets. Statist. Probab. Lett. 65 (2003) 411-417. | MR | Zbl

[3] A. Baíllo, J.A. Cuesta-Albertos and A. Cuevas, Convergence rates in nonparametric estimation of level sets. Statist. Probab. Lett. 53 (2001) 27-35. | MR | Zbl

[4] F. Belzunce, A. Castaño, A. Olvera-Cervantes and A. Suárez-Llorens, Quantile curves and dependence structure for bivariate distributions. Comput. Stat. Data Anal. 51 (2007) 5112-5129. | MR | Zbl

[5] G. Biau, B. Cadre and B. Pelletier, A graph-based estimator of the number of clusters. ESAIM : PS 11 (2007) 272-280. | Numdam | MR | Zbl

[6] 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). | MR | Zbl

[7] B. Cadre, Kernel estimation of density level sets. J. Multivar. Anal. 97 (2006) 999-1023. | MR | Zbl

[8] J. Cai and H. Li, Conditional tail expectations for multivariate phase-type distributions. J. Appl. Probab. 42 (2005) 810-825. | MR | Zbl

[9] L. Cavalier, Nonparametric estimation of regression level sets. Statistics (Berl. DDR) 29 (1997) 131-160. | MR | Zbl

[10] Y.P. Chaubey and P.K. Sen, 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. | MR | Zbl

[11] A. Cuevas and R. Fraiman, A plug-in approach to support estimation. Ann. Stat. 25 (1997) 2300-2312. | MR | Zbl

[12] A. Cuevas and A. Rodríguez-Casal, On boundary estimation. Adv. Appl. Probab. 36 (2004) 340-354. | MR | Zbl

[13] A. Cuevas, W. González-Manteiga and A. Rodríguez-Casal, Plug-in estimation of general level sets. Australian & New Zealand J. Statist. 48 (2006) 7-19. | MR | Zbl

[14] L. De Haan and X. Huang, Large quantile estimation in a multivariate setting. J. Multivar. Anal. 53 (1995) 247-263. | MR | Zbl

[15] S. Dedu and R. Ciumara, Restricted optimal retention in stop-loss reinsurance under VaR and CTE risk measures. Proc. of Rom. Acad. Ser. A 11 (2010) 213-217. | MR

[16] M. Denuit, J. Dhaene, M. Goovaerts and R. Kaas, Actuarial Theory for Dependent Risks. Wiley, (2005).

[17] P. Embrechts and G. Puccetti, Bounds for functions of multivariate risks. J. Multivar. Anal. 97 (2006) 526-547. | MR | Zbl

[18] J.M. Fernández-Ponce and A. Suárez-Lloréns, Central regions for bivariate distributions. Austrian J. Stat. 31 (2002) 141-156.

[19] E.W. Frees and E.A. Valdez, Understanding relationships using copulas. North Amer. Actuar. J. 2 (1998) 1-25. | MR | Zbl

[20] J.A. Hartigan, Estimation of a convex density contour in two dimensions. J. Amer. Statist. Assoc. 82 (1987) 267-270. | MR | Zbl

[21] V.I. Koltchinskii, M-estimation, convexity and quantiles. Ann. Statist. 25 (1997) 435-477. | MR | Zbl

[22] T. Laloë, Sur Quelques Problèmes d'Apprentissage Supervisé et Non Supervisé. Ph.D. thesis, University Montpellier II (2009).

[23] J.-C. Massé and R. Theodorescu, Halfplane trimming for bivariate distributions. J. Multivar. Anal. 48 (1994) 188-202. | MR | Zbl

[24] G. Nappo and F. Spizzichino, Kendall distributions and level sets in bivariate exchangeable survival models. Inform. Sci. 179 (2009) 2878-2890. | MR | Zbl

[25] W. Polonik, Measuring mass concentrations and estimating density contour clusters - an excess mass approach. Ann. Stat. 23 (1995) 855-881. | MR | Zbl

[26] W. Polonik, Minimum volume sets and generalized quantile processes. Stoch. Proc. Appl. 69 (1997) 1-24. | MR | Zbl

[27] P. Rigollet and R. Vert, Optimal rates for plug-in estimators of density level sets. Bernoulli. 15 (2009) 1154-1178. | MR | Zbl

[28] A. Rodríguez-Casal. Estimacíon de conjuntos y sus fronteras. Un enfoque geometrico. Ph.D. thesis, University of Santiago de Compostela (2003).

[29] C. Rossi, 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. | Zbl

[30] C. Rossi, Proprietà geometriche delle superficie di ripartizione. Rend. Mat. (6) 9 (1976) 725-736 (1977). | MR | Zbl

[31] R. Serfling, Quantile functions for multivariate analysis : approaches and applications. Stat. Neerlandica 56 (2002) 214-232 Special issue : Frontier research in theoretical statistics (2000) (Eindhoven). | MR | Zbl

[32] L. Tibiletti, On a new notion of multidimensional quantile. Metron 51 (1993) 77-83. | MR | Zbl

[33] A.B. Tsybakov, On nonparametric estimation of density level sets. Ann. Stat. 25 (1997) 948-969. | MR | Zbl

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