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Optimally robust estimators in generalized Pareto models. (English) Zbl 1440.62100

Summary: In this paper, we study the robustness properties of several procedures for the joint estimation of shape and scale in a generalized Pareto model. The estimators that we primarily focus upon, most bias robust estimator (MBRE) and optimal MSE-robust estimator (OMSE), are one-step estimators distinguished as optimally robust in the shrinking neighbourhood setting; that is, they minimize the maximal bias, respectively, on such a specific neighbourhood, the maximal mean squared error (MSE). For their initialization, we propose a particular location-dispersion estimator, MedkMAD, which matches the population median and kMAD (an asymmetric variant of the median of absolute deviations) against the empirical counterparts. These optimally robust estimators are compared to the maximum-likelihood, skipped maximum-likelihood, Cramér-von-Mises minimum distance, method-of-medians, and Pickands estimators. To quantify their deviation from robust optimality, for each of these suboptimal estimators, we determine the finite-sample breakdown point and the influence function, as well as the statistical accuracy measured by asymptotic bias, variance, and MSE – all evaluated uniformly on shrinking neighbourhoods. These asymptotic findings are complemented by an extensive simulation study to assess the finite-sample behaviour of the considered procedures. The applicability of the procedures and their stability against outliers are illustrated for the Danish fire insurance data set from the R package evir.

MSC:

62F35 Robustness and adaptive procedures (parametric inference)
62F10 Point estimation
62F12 Asymptotic properties of parametric estimators
62P05 Applications of statistics to actuarial sciences and financial mathematics

Software:

POT; R; ROptEst; RobASt; CRAN; RobLox; evir
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References:

[1] DOI: 10.1214/aop/1176996548 · Zbl 0295.60014 · doi:10.1214/aop/1176996548
[2] DOI: 10.1214/aos/1176343003 · Zbl 0312.62038 · doi:10.1214/aos/1176343003
[3] Basel Committee on Banking Supervision, International Convergence of Capital Measurement and Capital Standards: A Revised Framework (2006)
[4] Horbenko N., J. Oper. Risk 6 (2) pp 3– (2011) · doi:10.21314/JOP.2011.090
[5] Neslehova J., J. Oper. Risk 1 (1) pp 3– (2006) · doi:10.21314/JOP.2006.016
[6] DOI: 10.1080/01621459.1996.10476735 · doi:10.1080/01621459.1996.10476735
[7] DOI: 10.1023/A:1009975020370 · Zbl 0947.62034 · doi:10.1023/A:1009975020370
[8] Dupuis D. J., Extremes 1 (3) pp 251– (1998) · Zbl 0921.62030 · doi:10.1023/A:1009914915709
[9] Dupuis D. J., Canad. J. Statist 34 (4) pp 639– (2006) · Zbl 1115.62056 · doi:10.1002/cjs.5550340406
[10] Vandewalle B., Comput. Statist. Data Anal 51 (12) pp 6252– (2007) · Zbl 1445.62102 · doi:10.1016/j.csda.2007.01.003
[11] DOI: 10.1214/aos/1176343247 · Zbl 0323.62033 · doi:10.1214/aos/1176343247
[12] Tsourti Z., Statistics Tech (2001)
[13] DOI: 10.1214/aos/1176350499 · Zbl 0642.62022 · doi:10.1214/aos/1176350499
[14] DOI: 10.1017/CBO9780511802256 · doi:10.1017/CBO9780511802256
[15] Cope E. W., J. Oper. Risk 4 (4) pp 3– (2009) · doi:10.21314/JOP.2009.069
[16] Dupuis D. J., Canad. J. Statist 30 (1) pp 17– (2002) · Zbl 1003.62016 · doi:10.2307/3315863
[17] Hampel F. R., Robust Statistics. The Approach Based on Influence Functions (1986) · Zbl 0593.62027
[18] DOI: 10.1002/(SICI)1097-0258(19990815)18:15<1993::AID-SIM165>3.0.CO;2-H · doi:10.1002/(SICI)1097-0258(19990815)18:15<1993::AID-SIM165>3.0.CO;2-H
[19] Peng L., Extremes 4 (1) pp 53– (2001) · Zbl 1008.62024 · doi:10.1023/A:1012233423407
[20] DOI: 10.1080/01621459.1997.10473683 · doi:10.1080/01621459.1997.10473683
[21] DOI: 10.1080/00401706.1987.10488243 · doi:10.1080/00401706.1987.10488243
[22] DOI: 10.1080/03610929808832136 · Zbl 0900.62125 · doi:10.1080/03610929808832136
[23] DOI: 10.1111/j.1467-842X.2006.00464.x · Zbl 1117.62023 · doi:10.1111/j.1467-842X.2006.00464.x
[24] Brazauskas V., Insur. Math. Econ 45 (3) pp 424– (2009) · Zbl 1231.91148 · doi:10.1016/j.insmatheco.2009.09.002
[25] Juárez S. F., Extremes 7 (3) pp 237– (2004) · Zbl 1091.62017 · doi:10.1007/s10687-005-6475-6
[26] DOI: 10.1016/j.csda.2005.09.011 · Zbl 1157.62399 · doi:10.1016/j.csda.2005.09.011
[27] DOI: 10.1007/978-1-4684-0624-5 · doi:10.1007/978-1-4684-0624-5
[28] Ruckdeschel P., Metrika (2011)
[29] A. McNeil, (original in S), A. Stephenson (R port),evir: Extreme Values in R, package, version 1.6, 2008, software available at http://cran.r-project.org/.
[30] Huber-Carol, C. 1970. ”Étude asymptotique de tests robustes”. Zürich: ETH. Ph.D. diss
[31] DOI: 10.1214/aos/1176344312 · Zbl 0411.62020 · doi:10.1214/aos/1176344312
[32] Bickel P. J., Ecole d’Eté de Probabilités de Saint Flour IX 1979 pp 1– (1981) · Zbl 0455.00015 · doi:10.1007/BFb0097497
[33] Rieder H., Stat. Methods Appl 17 (1) pp 13– (2008) · Zbl 1367.62083 · doi:10.1007/s10260-007-0047-7
[34] Hampel, F. R. 1968. ”Contributions to the theory of robust estimation”. Berkeley: University of California. Ph.D. diss
[35] DOI: 10.1007/s10260-010-0133-0 · Zbl 1333.62095 · doi:10.1007/s10260-010-0133-0
[36] Fernholz L. T., Von Mises Calculus for Statistical Functionals (1979) · Zbl 0525.62031
[37] Kohl, M. 2005. ”Numerical contributions to the asymptotic theory of robustness”. Bayreuth: Universität. Ph.D. diss., Available athttp://stamats.de/ThesisMKohl.pdf · Zbl 1189.62051
[38] He X., Ann. Statist 33 (3) pp 998– (2005)
[39] DOI: 10.1524/stnd.22.3.201.57067 · Zbl 1057.62024 · doi:10.1524/stnd.22.3.201.57067
[40] DOI: 10.1016/S0167-9473(99)00018-3 · Zbl 04555904 · doi:10.1016/S0167-9473(99)00018-3
[41] DOI: 10.1080/01621459.1993.10476408 · doi:10.1080/01621459.1993.10476408
[42] Ruckdeschel P., Tech. Rep. No. 182 (2010)
[43] Field C., Int. Rev 62 (3) pp 405– (1994)
[44] DOI: 10.1016/S0167-9473(97)00011-X · Zbl 0900.62119 · doi:10.1016/S0167-9473(97)00011-X
[45] DOI: 10.1080/02331889808802657 · Zbl 1077.62513 · doi:10.1080/02331889808802657
[46] DOI: 10.1016/S0378-3758(02)00265-3 · Zbl 1178.62074 · doi:10.1016/S0378-3758(02)00265-3
[47] Development Core Team R, A Language and Environment for Statistical Computing (2009)
[48] M. Kohl and P. Ruckdeschel,ROptEst: Optimally robust estimation, Package available in version 0.8 on CRAN, 2009, software available at http://cran.r-project.org/.
[49] M. Ribatet,POT: Generalized Pareto distribution and peaks over threshold, package, version 1.1-0, 2009, software available at http://cran.r-project.org/.
[50] M. Kohl,RobLox: Optimally robust influence curves and estimators for location and scale, package available in version 0.8 on CRAN, 2009, software available at http://cran.r-project.org/.
[51] Hájek J., Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability 1 pp 175– (1972)
[52] DOI: 10.1002/0471725250 · Zbl 0536.62025 · doi:10.1002/0471725250
[53] Donoho D. L., Ann. Statist 16 (2) pp 552– (1988) · Zbl 0684.62030 · doi:10.1214/aos/1176350820
[54] DOI: 10.1016/j.spl.2009.05.001 · Zbl 1456.62056 · doi:10.1016/j.spl.2009.05.001
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