×

Estimating large losses in insurance analytics and operational risk using the g-and-h distribution. (English) Zbl 1479.91305

Summary: In this paper, we study the estimation of parameters for g-and-h distributions. These distributions find applications in modeling highly skewed and fat-tailed data, like extreme losses in the banking and insurance sector. We first introduce two estimation methods: a numerical maximum likelihood technique, and an indirect inference approach with a bootstrap weighting scheme. In a realistic simulation study, we show that indirect inference is computationally more efficient and provides better estimates than the maximum likelihood method in the case of extreme features in the data. Empirical illustrations on insurance and operational losses illustrate these findings.

MSC:

91G05 Actuarial mathematics
62P05 Applications of statistics to actuarial sciences and financial mathematics
62G32 Statistics of extreme values; tail inference

Software:

gk; QRM; kSamples; GAMLSS
PDFBibTeX XMLCite
Full Text: DOI Link

References:

[1] Bee, M. and Trapin, L., A simple approach to the estimation of Tukey’s gh distribution. J. Stat. Comput. Simul., 2016, 86(16), 3287-3302. doi: 10.1080/00949655.2016.1164159 · Zbl 07184798
[2] Bee, M., Hambuckers, J. and Trapin, L., Estimating value-at-risk for the g-and-h distribution: An indirect inference approach. Quant. Finance, 2019, 19, 1255-1266. doi: 10.1080/14697688.2019.1580762
[3] Beirlant, J., Dierckx, G., Goegebeur, Y. and Matthys, G., Tail index estimation and an exponential regression model. Extremes, 1999, 2(2), 177-200. doi: 10.1023/A:1009975020370 · Zbl 0947.62034 · doi:10.1023/A:1009975020370
[4] Calzolari, G., Fiorentini, G. and Sentana, E., Constrained indirect estimation. Rev. Econom. Stud., 2004, 71(4), 945-973. doi: 10.1111/0034-6527.00310 · Zbl 1060.62027 · doi:10.1111/0034-6527.00310
[5] Cruz, M., Peters, G. and Shevchenko, P., Fundamental Aspects of Operational Risk and Insurance Analytics: A Handbook of Operational Risk, 2015 (Wiley: Hoboken). · doi:10.1002/9781118573013
[6] Degen, M., Embrechts, P. and Lambrigger, D.D., The quantitative modeling of operational risk: Between g-and-h and EVT. Astin Bull., 2007, 37(2), 265-291. doi: 10.1017/S0515036100014860 · Zbl 1154.62077 · doi:10.1017/S0515036100014860
[7] Ding, B. and Shawky, H., The performance of hedge fund strategies and the asymmetry of return distributions. Euro. Financ. Manage., 2007, 13(2), 309-331. doi: 10.1111/j.1468-036X.2006.00356.x · doi:10.1111/j.1468-036X.2006.00356.x
[8] Dupuis, D. and Field, C., Large wind speeds: Modeling and outlier detection. J. Agric. Biol. Environ. Stat., 2004, 9(1), 105-121. doi: 10.1198/1085711043163 · doi:10.1198/1085711043163
[9] Dutta, K.K. and Perry, J., A tale of tails: An empirical analysis of loss distribution models for estimating operational risk capital. Technical Report 06-13, Federal Reserve Bank of Boston, 2006. · doi:10.2139/ssrn.918880
[10] Fernández, C. and Steel, M., On Bayesian modeling of fat tails and skewness. J. Am. Stat. Assoc., 1998, 93(441), 359-371. · Zbl 0910.62024
[11] Garcia, R., Renault, E. and Veredas, D., Estimation of stable distributions by indirect inference. J. Econom., 2011, 161(2), 325-337. doi: 10.1016/j.jeconom.2010.12.007 · Zbl 1441.62699 · doi:10.1016/j.jeconom.2010.12.007
[12] Gourieroux, C., Monfort, A. and Renault, E., Indirect inference. J. Appl. Econom. Suppl. Special Issue Econom. Inference Using Simulat. Techniq., 1993, 8, S85-S118. · Zbl 1448.62202
[13] Gouriéroux, C., Renault, E. and Touzi, N., Calibration by simulation for small sample bias correction. In Simulation-Based Inference in Econometrics: Methods and Applications, edited by R. Mariano, T. Schuermann, and M. J. Weeks, pp. 328-358, 2000 (Cambridge University Press: Cambridge). · Zbl 1184.62048 · doi:10.1017/CBO9780511751981.018
[14] Hambuckers, J., Groll, A. and Kneib, T., Understanding the economic determinants of the severity of operational losses: A regularized generalized pareto regression approach. J. Appl. Econom., 2018, 33(6), 898-935. doi: 10.1002/jae.2638 · doi:10.1002/jae.2638
[15] Headrick, T., Kowalchuk, R. and Sheng, Y., Parametric probability densities and distribution functions for Tukey g-and-h transformations and their use for fitting data. Appl. Math. Sci., 2008, 2, 449-462. · Zbl 1153.65010
[16] Hoaglin, D.C., Summarizing shape numerically: The g-and-h distributions. In Exploring Data Tables, Trends, and Shapes, pp. 461-513, 1985. · doi:10.1002/9781118150702.ch11
[17] Hodis, F., Headrick, T. and Sheng, Y., Power method distributions through conventional moments and L-moments. Appl. Math. Sci., 2012, 6(44), 2159-2193. · Zbl 1263.65003
[18] Huber, P.J., Robust Statistics, 1981 (Wiley: New York). · Zbl 0536.62025 · doi:10.1002/0471725250
[19] Jiang, W. and Turnbull, B., The indirect method: Inference based on intermediate statistics - A synthesis and examples. Stat. Sci., 2004, 19(2), 239-263. doi: 10.1214/088342304000000152 · Zbl 1100.62025 · doi:10.1214/088342304000000152
[20] McNeil, A., Frey, R. and Embrechts, P., Quantitative Risk Management: Concepts, Techniques, Tools, 2nd ed., 2015 (Princeton University Press: Princeton). · Zbl 1337.91003
[21] Moscadelli, M., The modelling of operational risk: Experiences with the analysis of the data collected by the basel committee. Technical report, Bank of Italy, Working Paper No 517, 2004. · doi:10.2139/ssrn.557214
[22] Peters, G.W. and Sisson, S., Bayesian inference, Monte Carlo sampling and operational risk. J. Oper. Risk, 2006, 1(3), 27-50. doi: 10.21314/JOP.2006.014 · doi:10.21314/JOP.2006.014
[23] Peters, G.W., Chen, W.Y. and Gerlach, R.H., Estimating quantile families of loss distributions for non-life insurance modelling via L-moments. Risks, 2016, 4(2), 14. doi: 10.3390/risks4020014 · doi:10.3390/risks4020014
[24] Prangle, D., gk: An R package for the g-and-k and generalised g-and-h distributions. ArXiv e-prints, 1706.06889v1, 2017.
[25] Rayner, G.D. and MacGillivray, H.L., Numerical maximum likelihood estimation for the g-and-k and generalized g-and-h distributions. Stat. Comput., 2002, 12(1), 57-75. doi: 10.1023/A:1013120305780 · Zbl 1247.62069 · doi:10.1023/A:1013120305780
[26] Rigby, R. and Stasinopoulos, D., Generalized additive models for location, scale and shape. J. R. Stat. Soc. Ser. C: Appl. Stat., 2005, 54(3), 507-554. doi: 10.1111/j.1467-9876.2005.00510.x · Zbl 1490.62201 · doi:10.1111/j.1467-9876.2005.00510.x
[27] Scholz, F. and Zhu, A., kSamples: K-sample rank tests and their combinations. R package version 1.2-9, 2019.
[28] Tukey, J.W., Modern techniques in data analysis. NSF-Sponsored Regional Research Conference at Southern Massachusetts University, North Dartmouth, 1977.
[29] Vogel, R. and Fennessey, N., L-moment diagrams should replace product moment diagrams. Water. Resour. Res., 1993, 29, 1745-1752. doi: 10.1029/93WR00341 · doi:10.1029/93WR00341
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.