Distortion risk measures, ROC curves, and distortion divergence

References

[1] D. Aldous and P. Diaconis, Strong uniform times and finite random walks, Adv. in Appl. Math. 8 (1987), 69–97. 10.1016/0196-8858(87)90006-6Search in Google Scholar

[2] B. C. Arnold, C. A. Robertson, P. L. Brockett and B.-Y. Shu, Generating ordered families of Lorenz curves by strongly unimodal distributions, J. Bus. Econom. Statist. 5 (1987), 305–308. 10.1080/07350015.1987.10509590Search in Google Scholar

[3] P. Artzner, F. Delbaen, J.-M. Eber and D. Heath, Coherent measures of risk, Math. Finance 9 (1999), 203–228. 10.1111/1467-9965.00068Search in Google Scholar

[4] H. Assa, On optimal reinsurance policy with distortion risk measures and premiums, Insurance Math. Econom. 61 (2015), 70–75. 10.1016/j.insmatheco.2014.11.007Search in Google Scholar

[5] A. G. Balter and A. Pelsser, Quantifying ambiguity bounds through hypothetical statistical testing, Working paper (2017), 10.2139/ssrn.2613843. 10.2139/ssrn.2613843Search in Google Scholar

[6] M. Basseville, Divergence measures for statistical data processing—an annotated bibliography, Signal Process. 93 (2013), 621–633. 10.1016/j.sigpro.2012.09.003Search in Google Scholar

[7] J. Belles-Sampera, J. M. Merigó, M. Guillén and M. Santolino, The connection between distortion risk measures and ordered weighted averaging operators, Insurance Math. Econom. 52 (2013), 411–420. 10.1016/j.insmatheco.2013.02.008Search in Google Scholar

[8] P. Billingsley, Probability and Measure, 2nd ed., Wiley, New York, 1986. Search in Google Scholar

[9] T. J. Boonen, Competitive equilibria with distortion risk measures, ASTIN Bull. 45 (2015), 703–728. 10.1017/asb.2015.11Search in Google Scholar

[10] S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press, Cambridge, 2004. 10.1017/CBO9780511804441Search in Google Scholar

[11] L. M. Bregman, The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming, Comput. Math. Math. Phys. 7 (1967), 200–217. 10.1016/0041-5553(67)90040-7Search in Google Scholar

[12] A. Cherny and D. Madan, New measures for performance evaluation, Rev. Financial Stud. 22 (2009), 2571–2606. 10.1093/rfs/hhn081Search in Google Scholar

[13] G. Choquet, Theory of capacities, Ann. Inst. Fourier 5 (1954), 131–295. 10.5802/aif.53Search in Google Scholar

[14] D. Denneberg, Non-Additive Measure and Integral, Kluwer, Dordrecht, 1994. 10.1007/978-94-017-2434-0Search in Google Scholar

[15] M. Denuit, J. Dhaene, M. Goovaerts and R. Kaas, Actuarial Theory for Dependent Risks: Measures, Orders and Models, Wiley, Chichester, 2005. 10.1002/0470016450Search in Google Scholar

[16] E. J. Dudewicz and S. Mishra, Modern Mathematical Statistics, Wiley, New York, 1988. Search in Google Scholar

[17] D. Ellsberg, Risk, ambiguity, and the Savage axioms, Quart. J. Econ. 75 (1961), 643–669. 10.2307/1884324Search in Google Scholar

[18] H. Föllmer and A. Schied, Stochastic Finance: An Introduction in Discrete Time, 3rd ed., Walter de Gruyter, Berlin, 2011. 10.1515/9783110218053Search in Google Scholar

[19] I. Gilboa and D. Schmeidler, Maxmin expected utility with non-unique prior, J. Math. Econom. 18 (1989), 141–153. 10.1016/0304-4068(89)90018-9Search in Google Scholar

[20] T. Goll and L. Rüschendorf, Minimax and minimal distance martingale measures and their relationship to portfolio optimization, Finance Stoch. 5 (2001), 557–581. 10.1007/s007800100052Search in Google Scholar

[21] M. Goovaerts, F. De Vylder and J. Haezendonck, Insurance Premiums, North-Holland, Amsterdam, 1984. 10.1007/978-94-009-6354-2Search in Google Scholar

[22] L. P. Hansen and T. J. Sargent, Robustness, Princeton University Press, Princeton, 2008. Search in Google Scholar

[23] W. Hürlimann, Distortion risk measures and economic capital, North Amer. Actuar. J. 8 (2004), 86–95. 10.1080/10920277.2004.10596130Search in Google Scholar

[24] W. Hürlimann, On the lookback distortion risk measure: Theory and applications, J. Math. Sci. Adv. Appl. 30 (2014), 21–47. Search in Google Scholar

[25] E. L. Lehmann and J. P. Romano, Testing Statistical Hypotheses, 3rd ed., Springer, New York, 2005. Search in Google Scholar

[26] K. Postek, D. den Hertog and B. Melenberg, Computationally tractable counterparts of distributionally robust constraints on risk measures, SIAM Rev. 58 (2016), 603–650. 10.1137/151005221Search in Google Scholar

[27] R. M. Reesor and D. L. McLeish, Risk, entropy, and the transformation of distributions, North Amer. Actuar. J. 7 (2003), no. 2, 128–144. Search in Google Scholar

[28] R. T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, 1997. Search in Google Scholar

[29] D. Schmeidler, Integral representation without additivity, Proc. Amer. Math. Soc. 97 (1986), 255–261. 10.1090/S0002-9939-1986-0835875-8Search in Google Scholar

[30] E. N. Sereda, E. M. Bronshtein, S. T. Rachev, F. J. Fabozzi, W. Sun and S. V. Stoyanov, Distortion risk measures in portfolio optimization, Handbook of Portfolio Construction. Contemporary Applications of Markowitz Techniques, Springer, New York (2010), 649–673. 10.1007/978-0-387-77439-8_25Search in Google Scholar

[31] A. Tsanakas, Dynamic capital allocation with distortion risk measures, Insurance Math. Econ. 35 (2004), 223–243. 10.1016/j.insmatheco.2003.09.005Search in Google Scholar

[32] H. Tsukahara, One-parameter families of distortion risk measures, Math. Finance 19 (2009), 691–705. 10.1111/j.1467-9965.2009.00385.xSearch in Google Scholar

[33] A. Wald, Sequential tests of statistical hypotheses, Ann. of Math. Stud. 16 (1945), 117–186. 10.1214/aoms/1177731118Search in Google Scholar

[34] S. S. Wang, Insurance pricing and increased limits ratemaking by proportional hazards transforms, Insurance Math. Econ. 17 (1995), 43–54. 10.1016/0167-6687(95)00010-PSearch in Google Scholar

[35] S. S. Wang, Premium calculation by transforming the layer premium density, ASTIN Bull. 26 (1996), 71–92. 10.2143/AST.26.1.563234Search in Google Scholar

[36] S. S. Wang, A class of distortion operators for pricing financial and insurance risks, J. Risk Insurance 67 (2000), 15–36. 10.2307/253675Search in Google Scholar

[37] S. S. Wang, V. R. Young and H. H. Panjer, Axiomatic characterization of insurance prices, Insurance Math. Econ. 21 (1997), 173–183. 10.1016/S0167-6687(97)00031-0Search in Google Scholar

[38] J. L. Wirch and M. R. Hardy, A synthesis of risk measures for capital adequacy, Insurance Math. Econ. 25 (1999), 337–347. 10.1016/S0167-6687(99)00036-0Search in Google Scholar

[39] V. R. Young, Optimal insurance under Wang’s premium principle, Insurance Math. Econ. 25 (1999), 109–122. 10.1016/S0167-6687(99)00012-8Search in Google Scholar

[40] Bank for International Settlements, Minimum capital requirements for market risk, 2016, www.bis.org/bcbs/publ/d352.pdf. Search in Google Scholar