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ROBUST AND EFFICIENT FITTING OF SEVERITY MODELS AND THE METHOD OF WINSORIZED MOMENTS

Published online by Cambridge University Press:  02 November 2017

Qian Zhao ,
Vytaras Brazauskas  and
Jugal Ghorai
Qian Zhao
Affiliation:
Department of Mathematics, Robert Morris University, Moon Township, PA 15108, USA, E-Mail: zhao@rmu.edu
Vytaras Brazauskas*
Affiliation:
Department of Mathematical Sciences, University of Wisconsin-Milwaukee, P.O. Box 413, Milwaukee, WI 53201, USA
Jugal Ghorai
Affiliation:
Department of Mathematical Sciences, University of Wisconsin-Milwaukee, P.O. Box 413, Milwaukee, WI 53201, USA, E-Mail: jugal@uwm.edu
*
E-Mail: vytaras@uwm.edu
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Abstract

Continuous parametric distributions are useful tools for modeling and pricing insurance risks, measuring income inequality in economics, investigating reliability of engineering systems, and in many other areas of application. In this paper, we propose and develop a new method for estimation of their parameters—the method of Winsorized moments (MWM)—which is conceptually similar to the method of trimmed moments (MTM) and thus is robust and computationally efficient. Both approaches yield explicit formulas of parameter estimators for log-location-scale families and their variants, which are commonly used to model claim severity. Large-sample properties of the new estimators are provided and corroborated through simulations. Their performance is also compared to that of MTM and the maximum likelihood estimators (MLE). In addition, the effect of model choice and parameter estimation method on risk pricing is illustrated using actual data that represent hurricane damages in the United States from 1925 to 1995. In particular, the estimated pure premiums for an insurance layer are computed when the lognormal and log-logistic models are fitted to the data using the MWM, MTM and MLE methods.

Keywords

Claim severityefficiencyinsurance layerrobustnesswinsorized data
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 48 , Issue 1 , January 2018 , pp. 275 - 309
Copyright
Copyright © Astin Bulletin 2017 

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References

Brazauskas, V. (2009) Robust and efficient fitting of loss models: Diagnostic tools and insights. North American Actuarial Journal, 13 (3), 1–14.CrossRefGoogle Scholar
Brazauskas, V., Jones, B.L. and Zitikis, R. (2007) Robustification and performance evaluation of empirical risk measures and other vector-valued estimators. Metron, LXV (2), 175–199.Google Scholar
Brazauskas, V., Jones, B. and Zitikis, R. (2009) Robust fitting of claim severity distributions and the method of trimmed moments. Journal of Statistical Planning and Inference, 139 (6), 2028–2043.Google Scholar
Brazauskas, V. and Kleefeld, A. (2009) Robust and efficient fitting of the generalized Pareto distribution with actuarial applications in view. Insurance: Mathematics and Economics, 45 (3), 424–435.Google Scholar
Brazauskas, V. and Kleefeld, A. (2011) Folded and log-folded-t distributions as models for insurance loss data. Scandinavian Actuarial Journal, 2011 (1), 59–79.CrossRefGoogle Scholar
Brazauskas, V. and Serfling, R. (2000) Robust and efficient estimation of the tail index of a single-parameter Pareto distribution (with discussion). North American Actuarial Journal, 4 (4), 12–27. Discussion: 5(3), 123–126. Reply: 5(3), 126–128.CrossRefGoogle Scholar
Brazauskas, V. and Serfling, R. (2003) Favorable estimators for fitting Pareto models: A study using goodness-of-fit measures with actual data. ASTIN Bulletin, 33 (2), 365–381.CrossRefGoogle Scholar
Chau, J. (2013) Robust Estimation In Operational Risk Modeling. M.S. Thesis. Utrecht University. Available at https://dspace.library.uu.nl/bitstream/handle/1874/290015/Thesis_Final.pdf. Accessed on October 1, 2017.Google Scholar
Chernoff, H., Gastwirth, J.L. and Jones, M.V. (1967) Asymptotic distribution of linear combinations of functions of order statistics with applications to estimation. The Annals of Mathematical Statistics, 38 (1), 52–72.Google Scholar
Cont, R. (2006) Model uncertainty and its impact on the pricing of derivative instruments. Mathematical Finance, 16 (3), 519–547.CrossRefGoogle Scholar
deCani, J.S. and Stine, R.A. (1986) A note on the information matrix for a logistic distribution. The American Statistician, 40 (3), 220–222.Google Scholar
Dell'Aquila, R. and Embrechts, P. (2006) Extremes and robustness: A contradiction? Financial Markets and Portfolio Management, 20 (1), 103–118.CrossRefGoogle Scholar
Dornheim, H. and Brazauskas, V. (2007) Robust and efficient methods for credibility when claims are approximately gamma-distributed. North American Actuarial Journal, 11 (3), 138–158.CrossRefGoogle Scholar
Dornheim, H. and Brazauskas, V. (2014) Case studies using credibility and corrected adaptively truncated likelihood methods. Variance, 7 (2), 168–192.Google Scholar
Gisler, A. and Reinhard, P. (1993) Robust credibility. ASTIN Bulletin, 23 (1), 118–143.CrossRefGoogle Scholar
Hansen, L.P. (1982) Large sample properties of generalized method of moments estimators. Econometrica, 50 (4), 1029–1054.Google Scholar
Hansen, L.P. and Sargent, T.J. (2008) Robustness. Princeton, NJ: Princeton University Press.Google Scholar
Horbenko, N., Ruckdeschel, P. and Bae, T. (2011) Robust estimation of operational risk. Journal of Operational Risk, 6 (2), 3–30.Google Scholar
Huber, P.J. and Ronchetti, E.M. (2009) Robust Statistics, 2nd edition. Hoboken, NJ: Wiley.CrossRefGoogle Scholar
Johnson, N.L., Kotz, S. and Balakrishnan, N. (1995) Continuous Univariate Distributions, Vol. 2, 2nd edition. New York: Wiley.Google Scholar
Kim, J.H.T. and Jeon, Y. (2013) Credibility theory based on trimming. Insurance: Mathematics and Economics, 53 (1), 36–47.Google Scholar
Kleefeld, A. and Brazauskas, V. (2012) A statistical application of the quantile mechanics approach: MTM estimators for the parameters of t and gamma distributions. European Journal of Applied Mathematics, 23 (5), 593–610.CrossRefGoogle Scholar
Kleiber, C. and Kotz, S. (2003) Statistical Size Distributions in Economics and Actuarial Sciences. New York: Wiley.Google Scholar
Klugman, S.A., Panjer, H.H. and Willmot, G.E. (2012) Loss Models: From Data to Decisions, 4th edition. New York: Wiley.Google Scholar
Ko, S-J. and Lee, Y.H. (1991) Theoretical analysis of Winsorizing smoothers and their applications to image processing. Proceedings of 1991 International Conference on Acoustics, Speech, and Signal Processing, 3001–3004. IEEE.Google Scholar
Künsch, H.R. (1992) Robust methods for credibility. ASTIN Bulletin, 22 (1), 33–49.Google Scholar
Marceau, E. and Rioux, J. (2001) On robustness in risk theory. Insurance: Mathematics and Economics, 29, 167–185.Google Scholar
McNeil, A.J., Frey, R. and Embrechts, P. (2005) Quantitative Risk Management: Concepts, Techniques and Tools. Princeton, NJ: Princeton University Press.Google Scholar
Nadarajah, S. and Bakar, S.A.A. (2015) New folded models for the log-transformed Norwegian fire claim data. Communications in Statistics: Theory and Methods, 44 (20), 4408–4440.CrossRefGoogle Scholar
Opdyke, J.D. and Cavallo, A. (2012) Estimating operational risk capital: The challenges of truncation, the hazards of MLE, and the promise of robust statistics. Journal of Operational Risk, 7 (3), 3–90.CrossRefGoogle Scholar
Pielke, R.A. Jr. and Landsea, C.W. (1998) Normalized hurricane damages in the United States: 1925–1995. Weather and Forecasting, 13, 621–631.2.0.CO;2>CrossRefGoogle Scholar
Serfling, R.J. (1980) Approximation Theorems of Mathematical Statistics. New York: Wiley.CrossRefGoogle Scholar
Serfling, R. (2002) Efficient and robust fitting of lognormal distributions (with discussion). North American Actuarial Journal, 6 (4), 95–109. Discussion: 7(3), 112–116. Reply: 7(3), 116.Google Scholar
Van Kerm, P. (2007) Extreme incomes and the estimation of poverty and inequality indicators from EU-SILC. IRISS Working Paper 2007-01, An Integrated Research Infrastructure in the Socio-Economic Sciences, 1–51.Google Scholar
Zhao, Q., Brazauskas, V. and Ghorai, J. (2017) Small-sample performance of the MTM and MWM estimators for the parameters of log-location-scale families. Submitted for publication.Google Scholar