Smoothed Quantiles for Measuring Discrete Risks

REFERENCES

• Acerbi, C., and D. Tasche. 2002. On the coherence of expected shortfall. Journal of Banking and Finance 26 (7):1487–503. doi:10.1016/S0378-4266(02)00283-2
   Web of Science ®Google Scholar
• Albrecht, P. 2004. Risk measures. In Encyclopedia of actuarial science, ed. B. Sundt and J. Teugels, vol. 3, 1493–501. London: Wiley.
   Google Scholar
• Acerbi, C., and D. Tasche.  2002. On the coherence of expected shortfall. Journal of Banking and Finance 26 (7):1487–503. doi:10.1016/S0378-4266(02)00283-2
   Web of Science ®Google Scholar
• Albrecht, P. 2004. Risk measures. In Encyclopedia of actuarial science, ed. B. Sundt and J. Teugels, vol. 3, 1493–501. London: Wiley.
   Google Scholar
• Alemany, R., C. Bolancé, and M. Guillén. 2013. A nonparametric approach to calculating value-at-risk. Insurance: Mathematics and Economics 52 (2):255–62. doi:10.1016/j.insmatheco.2012.12.008
   Web of Science ®Google Scholar
• Artzner, P., F. Delbaen, J.-M. Eber, and D. Heath. 1999. Coherent measures of risk. Mathematical Finance 9 (3):203–28. doi:10.1111/1467-9965.00068
   Web of Science ®Google Scholar
• Bolancé, C., and M. Guillén. 2021. Nonparametric estimation of extreme quantiles with an application to longevity risk. Risks 9 (77):1–23. doi:10.3390/risks9040077
   Google Scholar
• Brazauskas, V., B. Jones, M. Puri, and R. Zitikis. 2007. Nested L-statistics and their use in comparing the riskiness of portfolios. Scandinavian Actuarial Journal 107 (3):162–79. doi:10.1080/03461230701390287
   Google Scholar
• Brazauskas, V., B. Jones, M. Puri, and R. Zitikis. 2008. Estimating conditional tail expectations with actuarial applications in view. Journal of Statistical Planning and Inference 138 (11):3590–604. doi:10.1016/j.jspi.2005.11.011
   Web of Science ®Google Scholar
• Brazauskas, V., and T. Kaiser. 2004. Discussion of “Empirical estimation of risk measures and related quantities” by Jones and Zitikis. North American Actuarial Journal 8 (3):114–17. doi:10.1080/10920277.2004.10596155
   Google Scholar
• Furman, E., R. Wang, and R. Zitikis. 2017. Gini-type measures of risk and variability: Gini shortfall, capital allocations, and heavy-tailed risks. Journal of Banking and Finance 83:70–84. doi:10.1016/j.jbankfin.2017.06.013
   Web of Science ®Google Scholar
• Göb, R. 2011. Estimating value-at-risk and conditional value-at-risk for count variables. Quality and Reliability Engineering International 27:659–72. doi:10.1002/qre.1238
   Web of Science ®Google Scholar
• Harrell, F. E., and C. E. Davis. 1982. A new distribution-free quantile estimator. Biometrika 69 (3):635–40. doi:10.1093/biomet/69.3.635
   Web of Science ®Google Scholar
• Jones, B. L. and R. Zitikis (2003). Empirical estimation of risk measures and related quantities. North American Actuarial Journal, 7(4), 44–54.
   Google Scholar
• Jones, B. L., and R. Zitikis. 2005. Testing for the order of risk measures: An application of L-statistics in actuarial science. Metron 63 (2):193–211.
   Google Scholar
• Jones, B. L., and R. Zitikis. 2007. Risk measures, distortion parameters, and their empirical estimation. Insurance: Mathematics and Economics 41 (2):279–97. doi:10.1016/j.insmatheco.2006.11.001
   Web of Science ®Google Scholar
• Kabán, A. 2012. Non-parametric detection of meaningless distances in high dimensional data. Statistics and Computing 22 (2):375–85. doi:10.1007/s11222-011-9229-0
   Web of Science ®Google Scholar
• Kaiser, T., and V. Brazauskas. 2006. Interval estimation of actuarial risk measures. North American Actuarial Journal 10 (4):249–68. doi:10.1080/10920277.2006.10597425
   Google Scholar
• Klugman, S. A., H. H. Panjer, and G. E. Willmot. 2012. Loss models: From data to decisions. 4th ed. New York: Wiley.
   Google Scholar
• Ma, Y., M. G. Genton, and E. Parzen. 2011. Asymptotic properties of sample quantiles of discrete distributions. Annals of the Institute of Statistical Mathematics 63 (2):227–43. doi:10.1007/s10463-008-0215-z
   Web of Science ®Google Scholar
• Parzen, E. 1992. Unification of statistical methods for continuous and discrete data. In Computing science and statistics, ed. C. Page and R. LePage, 235–42. New York: Springer.
   Google Scholar
• Parzen, E. 2004. Quantile probability and statistical data modeling. Statistical Science 19 (4):652–62. doi:10.1214/088342304000000387
   Web of Science ®Google Scholar
• Ratnam, P. 2020. Smoothed quantiles for claim frequency models, with applications to risk measurement. Ph.D. diss., University of Wisconsin–Milwaukee.
   Google Scholar
• Samanthi, R., W. Wei, and V. Brazauskas. 2017. Comparing the riskiness of dependent portfolios via nested L-statistics. Annals of Actuarial Science 11 (2):237–52. doi:10.1017/S1748499516000233
   Google Scholar
• Saw, J. G., M. C. K. Yang, and T. C. Mo. 1984. Chebyshev inequality with estimated mean and variance. The American Statistician 38 (2):130–32.
   Web of Science ®Google Scholar
• Serfling, R. J. 1980. Approximation theorems of mathematical statistics. New York: Wiley.
   Google Scholar
• Stigler, S. M. 1977. Fractional order statistics, with applications. Journal of the American Statistical Association 72 (359):544–50. doi:10.1080/01621459.1977.10480611
   Web of Science ®Google Scholar
• Tapiero, C. S. 2004. Risk management: An interdisciplinary framework. In Encyclopedia of actuarial science, ed. B. Sundt and J. Teugels, vol. 3, 1483–93. London: Wiley.
   Google Scholar
• Wang, D., and A. D. Hutson. 2011. A fractional order statistic towards defining a smooth quantile function for discrete data. Journal of Statistical Planning and Inference 141 (9):3142–50. doi:10.1016/j.jspi.2011.03.035
   Web of Science ®Google Scholar
• Yip, K. C., and K. K. Yau. 2005. On modeling claim frequency data in general insurance with extra zeros. Insurance: Mathematics and Economics 36 (2):153–63. doi:10.1016/j.insmatheco.2004.11.002
   Web of Science ®Google Scholar