Abstract
In this paper, we present a procedure for consistent estimation of the severity and frequency distributions based on incomplete insurance data and demonstrate that ignoring the thresholds leads to a serious underestimation of the ruin probabilities. The event frequency is modelled with a non- homogeneous Poisson process with a sinusoidal intensity rate function. The choice of an adequate loss distribution is conducted via the in-sample goodness-of-fit procedures and forecasting, using classical and robust methodologies.
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Notes
1 For D and V statistics we use the scaling factor \(\sqrt{n}\), and for A2 and W2 we use n.
2 Results for the 25th, 50th and 75th percentiles are available in the original Working Paper, see https://doi.org/www.statistik.uni-karlsruhe.de/technical_reports/catastrophe.pdf.
3 Parameter estimates for the robust estimation approach and Figures for out-of-sample bootstrapped confidence intervals are available in the original Working Paper that can be downloaded from https://doi.org/www.statistik.uni-karlsruhe.de/technical_reports/catastrophe.pdf.
References
Burnecki K, Kukla G, Weron R (2000) Property insurance loss distributions. Physica A 287, 269–278
Burnecki K, Misiorek A, Weron R (2005). Loss distributions. In: Cizek P, Härdle W, Weron R (eds). Statistical tools for finance and insurance. Springer, Berlin Heidelberg New York, pp. 289–317
Burnecki K, Weron R (2005). Modeling of the risk process. In: Cizek P, Härdle W, Weron R (eds). Statistical tools for finance and insurance. Springer, Berlin Heidelberg New York, pp. 319–339
Chernobai A, Menn C, Trück S, Rachev S (2005a) A note on the estimation of the frequency and severity distribution of operational losses. Math Sci 30(2)
Chernobai A, Rachev S, Fabozzi F (2005b) Composite goodness-of-fit tests for left-truncated loss samples. Technical report, University of California Santa Barbara
Chernobai A, Trück S, Menn C, Rachev S (2005c) Estimation of operational Value-at-Risk in the presence of minimum collection thresholds. Technical report, University of California Santa Barbara
Grandell J (1991) Aspects of risk theory. Springer, Berlin Heidelberg New York
Hampel FR, Ronchetti EM, Rousseeuw RJ, Stahel WA (1986) Robust statistics: the approach based on influence functions. Wiley, New York
Huber PJ (2004) Robust statistics. Wiley, Hoboken
Klugman SA, Panjer HH, Willmot GE (1998) Loss models: from data to decisions. Wiley, New York
Knez PJ, Ready MJ (1997) On the robustness of size and book-to-market in cross-sectional regressions. J Financ 52:1355–1382
Martin RD, Simin TT (2003) Outlier resistant estimates of beta. Financ Anal J 59:56–69
Panjer H, Willmot G (1992) Insurance risk models. Society of Actuaries, Schaumburg
Patrik G (1981) Estimating casualty insurance loss amount distributions, In: Proceedings of the Casualty Actuarial Society. vol 67, pp 57–109
Rousseeuw PJ, Leroy AM (2003) Robust regression and outlier detection. Wiley, Hoboken
SwissRe (2004) Sigma preliminary report
Acknowledgment
We are thankful to S. Stoyanov of the FinAnalytica Inc. for partial computational assistance.
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Chernobai, A., Burnecki, K., Rachev, S. et al. Modelling catastrophe claims with left-truncated severity distributions. Computational Statistics 21, 537–555 (2006). https://doi.org/10.1007/s00180-006-0011-2
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DOI: https://doi.org/10.1007/s00180-006-0011-2