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Estimation in the Pareto Distribution

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  • Rytgaard, Mette

Abstract

In the present paper, different estimators of the Pareto parameter α will be proposed and compared to each others. First traditional estimators of α as the maximum likelihood estimator and the moment estimator will be deduced and their statistical properties will be analyzed. It is shown that the maximum likelihood estimator is biased but it can easily be modified to an minimum-variance unbiased estimator of a. But still the coefficient of variance of this estimator is very large. For similar portfolios containing same types of risks we will expect the estimated α-values to be at the same level. Therefore, credibility theory is used to obtain an alternative estimator of α which will be more stable and less sensitive to random fluctuations in the observed losses. Finally, an estimator of the risk premium for an unlimited excess of loss cover will be proposed. It is shown that this estimator is a minimum-variance unbiased estimator of the risk premium. This estimator of the risk premium will be compared to the more traditional methods of calculating the risk premium.

Suggested Citation

  • Rytgaard, Mette, 1990. "Estimation in the Pareto Distribution," ASTIN Bulletin, Cambridge University Press, vol. 20(2), pages 201-216, November.
    • Handle: RePEc:cup:astinb:v:20:y:1990:i:02:p:201-216_00
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    Cited by:

    1. Kenneth Gillingham & William D. Nordhaus & David Anthoff & Geoffrey Blanford & Valentina Bosetti & Peter Christensen & Haewon McJeon & John Reilly & Paul Sztorc, 2015. "Modeling Uncertainty in Climate Change: A Multi-Model Comparison," NBER Working Papers 21637, National Bureau of Economic Research, Inc.
    2. Arthur Charpentier & Emmanuel Flachaire, 2022. "Pareto models for top incomes and wealth," The Journal of Economic Inequality, Springer;Society for the Study of Economic Inequality, vol. 20(1), pages 1-25, March.
    3. Zema, Sebastiano Michele, 2022. "Uncovering the network structure of non-centrally cleared derivative markets: evidences from regulatory data," Working Paper Series 2721, European Central Bank.
    4. Sebastiano Michele Zema, 2023. "Uncovering the network structure of non-centrally cleared derivative markets: evidence from large regulatory data," Empirical Economics, Springer, vol. 65(4), pages 1799-1822, October.
    5. Pai, Jeffrey S., 1997. "Bayesian analysis of compound loss distributions," Journal of Econometrics, Elsevier, vol. 79(1), pages 129-146, July.
    6. Philip Vermeulen, 2018. "How Fat is the Top Tail of the Wealth Distribution?," Review of Income and Wealth, International Association for Research in Income and Wealth, vol. 64(2), pages 357-387, June.
    7. Hans Buhlmann & Pavel V. Shevchenko & Mario V. Wuthrich, 2009. "A "Toy" Model for Operational Risk Quantification using Credibility Theory," Papers 0904.1772, arXiv.org.
    8. Arthur Charpentier & Emmanuel Flachaire, 2019. "Pareto Models for Top Incomes," Working Papers hal-02145024, HAL.
    9. Mohamed E. Ghitany & Emilio Gómez-Déniz & Saralees Nadarajah, 2018. "A New Generalization of the Pareto Distribution and Its Application to Insurance Data," JRFM, MDPI, vol. 11(1), pages 1-14, February.
    10. Yogesh Tripathi & Somesh Kumar & Constantinos Petropoulos, 2016. "Estimating the shape parameter of a Pareto distribution under restrictions," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 79(1), pages 91-111, January.
    11. Safari, Muhammad Aslam Mohd & Masseran, Nurulkamal & Ibrahim, Kamarulzaman, 2018. "Optimal threshold for Pareto tail modelling in the presence of outliers," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 509(C), pages 169-180.
    12. Walid Abu-Dayyeh & Aissa Assrhani & Kamarulzaman Ibrahim, 2013. "Estimation of the shape and scale parameters of Pareto distribution using ranked set sampling," Statistical Papers, Springer, vol. 54(1), pages 207-225, February.
    13. Elbatal Ibrahim & Merovci Faton, 2014. "A Note on a Generalization of the Exponentiated Pareto Distribution," Stochastics and Quality Control, De Gruyter, vol. 29(1), pages 1-11, June.
    14. Milan Stehlík & Rastislav Potocký & Helmut Waldl & Zdeněk Fabián, 2010. "On the favorable estimation for fitting heavy tailed data," Computational Statistics, Springer, vol. 25(3), pages 485-503, September.
    15. Kim, Joseph H.T. & Jeon, Yongho, 2013. "Credibility theory based on trimming," Insurance: Mathematics and Economics, Elsevier, vol. 53(1), pages 36-47.

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