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  • Qian Zhao (a1), Vytaras Brazauskas (a2) and Jugal Ghorai (a3)


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.


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Brazauskas, V. (2009) Robust and efficient fitting of loss models: Diagnostic tools and insights. North American Actuarial Journal, 13 (3), 114.
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), 175199.
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), 20282043.
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), 424435.
Brazauskas, V. and Kleefeld, A. (2011) Folded and log-folded-t distributions as models for insurance loss data. Scandinavian Actuarial Journal, 2011 (1), 5979.
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), 1227. Discussion: 5(3), 123–126. Reply: 5(3), 126–128.
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), 365381.
Chau, J. (2013) Robust Estimation In Operational Risk Modeling. M.S. Thesis. Utrecht University. Available at Accessed on October 1, 2017.
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), 5272.
Cont, R. (2006) Model uncertainty and its impact on the pricing of derivative instruments. Mathematical Finance, 16 (3), 519547.
deCani, J.S. and Stine, R.A. (1986) A note on the information matrix for a logistic distribution. The American Statistician, 40 (3), 220222.
Dell'Aquila, R. and Embrechts, P. (2006) Extremes and robustness: A contradiction? Financial Markets and Portfolio Management, 20 (1), 103118.
Dornheim, H. and Brazauskas, V. (2007) Robust and efficient methods for credibility when claims are approximately gamma-distributed. North American Actuarial Journal, 11 (3), 138158.
Dornheim, H. and Brazauskas, V. (2014) Case studies using credibility and corrected adaptively truncated likelihood methods. Variance, 7 (2), 168192.
Gisler, A. and Reinhard, P. (1993) Robust credibility. ASTIN Bulletin, 23 (1), 118143.
Hansen, L.P. (1982) Large sample properties of generalized method of moments estimators. Econometrica, 50 (4), 10291054.
Hansen, L.P. and Sargent, T.J. (2008) Robustness. Princeton, NJ: Princeton University Press.
Horbenko, N., Ruckdeschel, P. and Bae, T. (2011) Robust estimation of operational risk. Journal of Operational Risk, 6 (2), 330.
Huber, P.J. and Ronchetti, E.M. (2009) Robust Statistics, 2nd edition. Hoboken, NJ: Wiley.
Johnson, N.L., Kotz, S. and Balakrishnan, N. (1995) Continuous Univariate Distributions, Vol. 2, 2nd edition. New York: Wiley.
Kim, J.H.T. and Jeon, Y. (2013) Credibility theory based on trimming. Insurance: Mathematics and Economics, 53 (1), 3647.
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), 593610.
Kleiber, C. and Kotz, S. (2003) Statistical Size Distributions in Economics and Actuarial Sciences. New York: Wiley.
Klugman, S.A., Panjer, H.H. and Willmot, G.E. (2012) Loss Models: From Data to Decisions, 4th edition. New York: Wiley.
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.
Künsch, H.R. (1992) Robust methods for credibility. ASTIN Bulletin, 22 (1), 3349.
Marceau, E. and Rioux, J. (2001) On robustness in risk theory. Insurance: Mathematics and Economics, 29, 167185.
McNeil, A.J., Frey, R. and Embrechts, P. (2005) Quantitative Risk Management: Concepts, Techniques and Tools. Princeton, NJ: Princeton University Press.
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), 44084440.
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), 390.
Pielke, R.A. Jr. and Landsea, C.W. (1998) Normalized hurricane damages in the United States: 1925–1995. Weather and Forecasting, 13, 621631.
Serfling, R.J. (1980) Approximation Theorems of Mathematical Statistics. New York: Wiley.
Serfling, R. (2002) Efficient and robust fitting of lognormal distributions (with discussion). North American Actuarial Journal, 6 (4), 95109. Discussion: 7(3), 112–116. Reply: 7(3), 116.
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.
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.
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ASTIN Bulletin: The Journal of the IAA
  • ISSN: 0515-0361
  • EISSN: 1783-1350
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