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Credible Means are exact Bayesian for Exponential Families

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  • Jewell, William S.

Abstract

The credibility formula used in casualty insurance experience rating is known to be exact for certain prior-likelihood distributions, and is the minimum least-squares unbiased estimator for all others. We show that credibility is, in fact, exact for all simple exponential families where the mean is the sufficient statistic, and is also exact in an extended sense for all regular distributions with their natural conjugate priors where there is a fixed-dimensional sufficient statistic.

Suggested Citation

  • Jewell, William S., 1974. "Credible Means are exact Bayesian for Exponential Families," ASTIN Bulletin, Cambridge University Press, vol. 8(1), pages 77-90, September.
    • Handle: RePEc:cup:astinb:v:8:y:1974:i:01:p:77-90_00
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    Cited by:

    1. Gómez-Déniz, E., 2008. "A generalization of the credibility theory obtained by using the weighted balanced loss function," Insurance: Mathematics and Economics, Elsevier, vol. 42(2), pages 850-854, April.
    2. Yang Lu, 2018. "Dynamic Frailty Count Process in Insurance: A Unified Framework for Estimation, Pricing, and Forecasting," Journal of Risk & Insurance, The American Risk and Insurance Association, vol. 85(4), pages 1083-1102, December.
    3. Gigante, Patrizia & Picech, Liviana & Sigalotti, Luciano, 2013. "Claims reserving in the hierarchical generalized linear model framework," Insurance: Mathematics and Economics, Elsevier, vol. 52(2), pages 381-390.
    4. Landsman, Zinoviy, 2002. "Credibility theory: a new view from the theory of second order optimal statistics," Insurance: Mathematics and Economics, Elsevier, vol. 30(3), pages 351-362, June.
    5. Bennett, Daniel & Chiang, Chun-Fang & Malani, Anup, 2015. "Learning during a crisis: The SARS epidemic in Taiwan," Journal of Development Economics, Elsevier, vol. 112(C), pages 1-18.
    6. Landsman, Zinoviy & Makov, Udi E., 1999. "Credibility evaluation for the exponential dispersion family," Insurance: Mathematics and Economics, Elsevier, vol. 24(1-2), pages 23-29, March.
    7. Jeong Himchan & Valdez Emiliano A., 2020. "Bayesian credibility premium with GB2 copulas," Dependence Modeling, De Gruyter, vol. 8(1), pages 157-171, January.
    8. Georgios Pitselis, 2024. "Credibility Distribution Estimation with Weighted or Grouped Observations," Risks, MDPI, vol. 12(1), pages 1-27, January.
    9. Jeong Himchan & Valdez Emiliano A., 2020. "Bayesian credibility premium with GB2 copulas," Dependence Modeling, De Gruyter, vol. 8(1), pages 157-171, January.
    10. Alicja Wolny-Dominiak & Tomasz Żądło, 2021. "The Measures of Accuracy of Claim Frequency Credibility Predictor," Sustainability, MDPI, vol. 13(21), pages 1-13, October.
    11. Emilio Gómez-Déniz & Enrique Calderín-Ojeda, 2018. "Multivariate Credibility in Bonus-Malus Systems Distinguishing between Different Types of Claims," Risks, MDPI, vol. 6(2), pages 1-11, April.
    12. Xacur, Oscar Alberto Quijano & Garrido, José, 2018. "Bayesian credibility for GLMs," Insurance: Mathematics and Economics, Elsevier, vol. 83(C), pages 180-189.
    13. Pitselis, Georgios, 2013. "Quantile credibility models," Insurance: Mathematics and Economics, Elsevier, vol. 52(3), pages 477-489.
    14. Pérez-Hornero, Patricia & Arias-Nicolás, José Pablo & Pulgarín, Antonio A. & Pulgarín, Antonio, 2013. "An annual JCR impact factor calculation based on Bayesian credibility formulas," Journal of Informetrics, Elsevier, vol. 7(1), pages 1-9.
    15. Marina Maniati & Evangelos Sambracos, 2017. "Decision-making process in shipping finance: A stochastic approach," Cogent Economics & Finance, Taylor & Francis Journals, vol. 5(1), pages 1317083-131, January.
    16. Paulsen, Jostein & Lunde, Astrid & Skaug, Hans Julius, 2008. "Fitting mixed-effects models when data are left truncated," Insurance: Mathematics and Economics, Elsevier, vol. 43(1), pages 121-133, August.
    17. Sebastian Calcetero-Vanegas & Andrei L. Badescu & X. Sheldon Lin, 2022. "Effective a Posteriori Ratemaking with Large Insurance Portfolios via Surrogate Modeling," Papers 2211.06568, arXiv.org, revised May 2023.
    18. Gómez-Déniz, E., 2018. "Adding a parameter to the exponential and Weibull distributions with applications," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 144(C), pages 108-119.
    19. Landsman, Zinoviy & Makov, Udi E., 2000. "On credibility evaluation and the tail area of the exponential dispersion family," Insurance: Mathematics and Economics, Elsevier, vol. 27(3), pages 277-283, December.
    20. Kim, Joseph H.T. & Jeon, Yongho, 2013. "Credibility theory based on trimming," Insurance: Mathematics and Economics, Elsevier, vol. 53(1), pages 36-47.
    21. Pitselis, Georgios, 2013. "Pure robust versus robust portfolio unbiased—Credibility and asymptotic optimality," Insurance: Mathematics and Economics, Elsevier, vol. 52(2), pages 391-403.
    22. Sánchez-Sánchez, M. & Sordo, M.A. & Suárez-Llorens, A. & Gómez-Déniz, E., 2019. "Deriving Robust Bayesian Premiums Under Bands Of Prior Distributions With Applications," ASTIN Bulletin, Cambridge University Press, vol. 49(1), pages 147-168, January.

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