Statistical Analysis of Stable Distribution Application in Non Life İnsurance
https://doi.org/10.26794/2587-5671-2024-28-5-146-155
Abstract
In recent years, the theory of stable variables has seen many exciting developments, due to the fact that it is a very rich class of probability laws able to represent different asymmetries, and heavy tails, so modelling complex phenomena; unlike normal law, which very often underestimates extreme events. α-stable distributions are a class of heavy-tailed distributions. For that, we will start in this paper by presenting a review of graphical tests, which will help us to verify if we are in the presence of data with infinite variance or not, and more precisely of stable distribution. Then we will apply these tests to real data representing car claim amounts, allowing us to assume that our sample follows a stable distribution. In order to confirm this hypothesis, we will therefore estimate the four parameters of the distribution using the McCuloch method, as well as the Koutrouvelis method in order to be able to make the diagnosis with Kernel Densities, and finally we will demonstrate that α-stable distribution is better fitted to the car claim amount data by using the Kolmogorov test.
About the Authors
A. Laouar
University of science and technology Houari Boumedienne (USTHB)
Algeria
Competing Interests:
Algeria
Amel Laouar, PhD student in Pure and Applied Mathematics, Research Associate
Laboratory of stochastic modelization and data mining
Algiers
Competing Interests:
The authors have no conflicts of interest to declare
K. Boukhetala
University of science and technology Houari Boumedienne (USTHB)
Algeria
Competing Interests:
Algeria
Kamal Boukhetala, PhD in Mathematics, Prof., Project manager and head of the team
Laboratory of Stochastic modelization and data mining; actuarial risk modeling-simulation team
Algiers
Competing Interests:
The authors have no conflicts of interest to declare
R. Sabre
National Higher Institute of Agronomic, Food and Environmental Sciences, AgroSup Dijon
France
Competing Interests:
France
Rachid Sabre, PhD in Mathematics
Dijon
Competing Interests:
The authors have no conflicts of interest to declare
References
1. Lévy P. Calcul des probabilités. Paris: Gauthier-Villars; 1925. 368 p.
2. Brookes B. C. Limit distributions for sums of independent random variables. By B. V. Gnedenko and A. N. Kolmogorov. Translated by K. L. Chung. Pp. ix, 264. $ 7.50. 1954. (Addison-Wesley, Cambridge, Mass.). The Mathematical Gazette. 1955;39(330):342–343. DOI: 10.2307/3608621
3. Werner T., Upper C. Time variation in the tail behaviour of bund futures returns. European Central Bank. Working Paper Series. 2002;(199). URL: https://www.ecb.europa.eu/pub/pdf/scpwps/ecbwp199.pdf (accessed on 15. 06. 2021).
4. Bamberg G., Dorfleitner G. Fat tails and traditional capital market theory. Institut für Statistik und Mathematische Wirtschaftstheorie, Universität Augsburg. 2001;(177). URL: https://www.yumpu.com/en/document/read/36398518/fat-tails-and-traditional-capital-market-theory-gilles-daniel
5. Zolotarev V. One-dimensional stable distributions. Providence, RI: American Mathematical Society; 1986. 284 p. (Translations of Mathematical Monographs. Vol. 65). DOI: 10.1090/mmono/065
6. Nolan J. P. Univariate stable distributions: Models for heavy tailed data. Cham: Springer-Verlag; 2020. 333 p. (Springer Series in Operations Research and Financial Engineering). DOI: 10.1007/978–3–030–52915–4
7. Nolan J. P. Maximum likelihood estimation and diagnostics for stable distributions. In: Barndorff-Nielsen O.E., Resnick S. I., Mikosch T., eds. Lévy processes. Boston, MA: Birkhäuser; 2001:379–400. DOI: 10.1007/978–1–4612–0197–7_17
8. Rolski T., Schmidli H., Schmidt V., Teugels J. Stochastic processes for insurance and finance. Hoboken, NJ: John Wiley & Sons, Inc.; 1999. 680 p. (Wiley Series in Probability and Statistics).
9. d’Estampes L. Traitement statistique des processus alpha-stables: mesures de dépendance et identification des ar stables. Test séquentiels tronqués. Thèse. Toulouse: Institut national polytechnique de Toulouse; 2003. 143 p. URL: https://tel.archives-ouvertes.fr/tel-00005216 (accessed on 15. 06. 2021).
10. Chambers J. M., Mallows C. L., Stuck B. W. A method for simulating stable random variables. Journal of the American Statistical Association. 1976;71(354):340–344. DOI: 10.1080/01621459.1976.10480344
11. Janicki A., Weron A. Simulation and chaotic behavior of alpha-stable stochastic processes. New York, NY: Marcel Dekker, Inc.; 1994 376 p. (Monographs and Textbooks in Pure and Applied Mathematics. Vol. 178).
12. Cramér H. Collective risk theory: A survey of the theory from the point of view of the theory of stochastic processes. Stockholm: Nordiska bokhandeln; 1955. 92 p.
13. McCulloch J. H. Simple consistent estimators of stable distribution parameters. Communications in Statistics — Simulation and Computation. 1986;15(4):1109–1136. DOI: 10.1080/03610918608812563
14. Fama E. F., Roll R. Parameter estimates for symmetric stable distributions. Journal of the American Statistical Association. 1971;66(334):331–338. DOI: 10.1080/01621459.1971.10482264
15. Reiss R.-D., Thomas M. Statistical analysis of extreme values: With applications to insurance, finance, hydrology and other fields. 3rd ed. Basel: Birkhäuser; 2007. 511 p.
16. Čížek P., Härdle W. K., Weron R., eds. Statistical tools for finance and insurance. Berlin, Heidelberg: Springer-Verlag; 2011. 420 p. DOI: 10.1007/978–3–642–18062–0
17. Koutrouvelis I. A. Regression-type estimation of the parameters of stable laws. Journal of the American Statistical Association. 1980;75(372):918–928. DOI: 10.1080/01621459.1980.10477573
18. Veillette M. Alpha-stable distributions for MATLAB. URL: http://math.bu.edu/people/mveillet/html/alphastablepub.html (accessed on 16. 06. 2021).
Review
For citations:
Laouar A., Boukhetala K., Sabre R. Statistical Analysis of Stable Distribution Application in Non Life İnsurance. Finance: Theory and Practice. 2024;28(5):146-155. https://doi.org/10.26794/2587-5671-2024-28-5-146-155
Views:
261
Email this article 