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DERIVING ROBUST BAYESIAN PREMIUMS UNDER BANDS OF PRIOR DISTRIBUTIONS WITH APPLICATIONS

Published online by Cambridge University Press:  23 November 2018

M. Sánchez-Sánchez ,
M.A. Sordo ,
A. Suárez-Llorens  and
E. Gómez-Déniz
M. Sánchez-Sánchez
Affiliation:
Departamento de Estadística e Investigación Operativa, Facultad de Ciencias, Universidad de Cádiz, Avda. Rep. Saharaui s/n, 11510 Puerto Real, Cádiz, Spain E-Mail: [email protected]
M.A. Sordo
Affiliation:
Departamento de Estadística e Investigación Operativa, Facultad de Ciencias, Universidad de Cádiz, Avda. Rep. Saharaui s/n, 11510 Puerto Real, Cádiz, Spain E-Mail: [email protected]
A. Suárez-Llorens*
Affiliation:
Departamento de Estadística e Investigación Operativa, Facultad de Ciencias, Universidad de Cádiz, Avda. Rep. Saharaui s/n, 11510 Puerto Real, Cádiz, Spain E-Mail: [email protected]
E. Gómez-Déniz
Affiliation:
Department of Quantitative MethodsUniversity of Las Palmas de Gran CanariaCampus Universitario de Tafira 35017 las Palmas de Gran Canaria, Spain E-Mail: [email protected]
*
E-Mail: [email protected]
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Abstract

We study the propagation of uncertainty from a class of priors introduced by Arias-Nicolás et al. [(2016) Bayesian Analysis, 11(4), 1107–1136] to the premiums (both the collective and the Bayesian), for a wide family of premium principles (specifically, those that preserve the likelihood ratio order). The class under study reflects the prior uncertainty using distortion functions and fulfills some desirable requirements: elicitation is easy, the prior uncertainty can be measured by different metrics, and the range of quantities of interest is easily obtained from the extremal members of the class. We illustrate the methodology with several examples based on different claim counts models.

Keywords

Credibilityclass of priorsdistortion functionsKolmogorov and Kantorovich metricspremium calculation principlerobust Bayesian analysisstochastic orders
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 49 , Issue 1 , January 2019 , pp. 147 - 168
Copyright
Copyright © Astin Bulletin 2018 

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References

Arias-Nicolás, J.P., Ruggeri, F. and Suárez-Llorens, A. (2016) New classes of priors based on stochastic orders and distortion functions. Bayesian Analysis, 11(4), 1107–1136.10.1214/15-BA984CrossRefGoogle Scholar
Bartoszewicz, J. and Skolimowska, M. (2006) Preservation of classes of life distributions and stochastics orders under weighting. Statistics & Probability Letters, 76, 587–59610.1016/j.spl.2005.09.003CrossRefGoogle Scholar
Belzunce, F., Martínez-Riquelme, C. and Mulero, J. (2016) An Introduction to Stochastics Orders. New York: Elsevier.Google Scholar
Berger, J. (1994) An overview of robust Bayesian analysis (with discussion). Test, 3, 5–124.10.1007/BF02562676CrossRefGoogle Scholar
Bickel, P.J. and Lehmann, E.L. (1979) Descriptive statistics for nonparametric models IV. Spread. In Contributions to Statistics (ed. Jureckova, J.). Dordrecht: Reidel.Google Scholar
Blazej, P. (2008) Preservation of classes of life distributions under weighting with a general weight function. Statistics & Probability Letters, 78, 3056–3061.10.1016/j.spl.2008.05.028CrossRefGoogle Scholar
Boratyǹska, A. (2017) Robust Bayesian estimation and prediction of reserves in exponential model with quadratic variance function. Insurance: Mathematics and Economics, 76, 135–140.Google Scholar
Bühlmann, H. (1967) Experience rating and credibility. ASTIN Bulletin, 4(3), 199–207.10.1017/S0515036100008989CrossRefGoogle Scholar
Bühlmann, H. (1980) An economic premium principle. ASTIN Bulletin, 11, 52–60.10.1017/S0515036100006619CrossRefGoogle Scholar
Bühlmann, H. and Gisler, A. (2005) A Course in Credibility Theory and Its Applications. Berlin: Springer.Google Scholar
Calderín, E. and Gómez-Déniz, E. (2007) Bayesian local robustness under weighted squared–error loss function incorporating unimodality. Statistics & Probability Letters, 77, 69–74.Google Scholar
Chan, J.S.K., Choy, S.T. and Makov, U.E. (2008) Robust Bayesian analysis of loss reserves data using the generalized-t distribution. ASTIN Bulletin, 38(1), 207–230.10.1017/S0515036100015142CrossRefGoogle Scholar
Denneberg, D. (1990) Premium calculation: Why standard deviation should be replaced by absolute deviation. ASTIN Bulletin, 20, 181–190.10.2143/AST.20.2.2005441CrossRefGoogle Scholar
Denuit, M., Dhaene, J., Goovaerts, M.J. and Kaas, K. 2005. Actuarial Theory for Dependent Risks. New York: John Wiley & Sons.10.1002/0470016450CrossRefGoogle Scholar
Drozdenko, V. (2008) Premium Calculations in Insurance. Actuarial Approach. Saarbrücken: VDM Verlag Dr. Müller.Google Scholar
Eichenauer, J., Lehn, J. and Rettig, S. (1988) A gamma-minimax result in credibility theory. Insurance: Mathematics and Economics 7, 49–57.Google Scholar
Furman, E. and Landsman, Z. (2006) On some risk-adjusted tail-based premium calculation principles. Journal of Actuarial Practice, 13, 175–190.Google Scholar
Furman, E. and Zitikis, R. (2008) Weighted premium calculation principle. Insurance: Mathematics and Economics, 42, 459–465.Google Scholar
Gerber, H.U. (1980) Credibility for Esscher premiums. Mitteilungen der VSVM, 80, 307–312.Google Scholar
Gómez-Déniz, E. (2009) Some Bayesian credibility premiums obtained by using posterior regret γ -minimax methodology. Bayesian Analysis, 4(2), 223–242.10.1214/09-BA408CrossRefGoogle Scholar
Gómez-Déniz, E., Hernández, A. and Vázquez-Polo, F.J. (1999) The Esscher premium principle in risk theory: A Bayesian sensitivity study. Insurance: Mathematics and Economics, 25, 387–395.Google Scholar
Gómez-Déniz, E., Hernández, A. and Vázquez-Polo, F.J. (2000) Robust Bayesian premium principles in actuarial science. Journal of the Royal Statistical Society (The Statistician, Series D), 49(2), 241–252.10.1111/1467-9884.00234CrossRefGoogle Scholar
Gómez-Déniz, E., Pérez, J.M., Hernández, A. and Vázquez-Polo, F.J. (2002) Measuring sensitivity in a bonus–malus system. Insurance: Mathematics and Economics, 31(1), 105–113.Google Scholar
Heilmann, W. (1989) Decision theoretic foundations of credibility theory. Insurance: Mathematics and Economics, 8(1), 75–95.Google Scholar
Heilmann, W. and Schroter, K. (1987) On the robustness of premium principles. Insurance: Mathematics and Economics, 6(2), 145–149.Google Scholar
Jewell, W.S. (1974) Credible means are exact Bayesian for exponential families. ASTIN Bulletin, 8(1), 77–90.10.1017/S0515036100009193CrossRefGoogle Scholar
Klugman, S., Panjer, H. and Willmot, G. (1998) Loss Model from Data to Decisions. New York: John Wiley & Sons.Google Scholar
Lau, J.W., Siu, T.K. and Yang, H. (2006) On Bayesian mixture credibility. ASTIN Bulletin, 36(2), 573–588.10.1017/S0515036100014677CrossRefGoogle Scholar
López-Díaz, M., Sordo, M.A. and Suárez-Llorens, A. (2012) On the lp metric between a probability distribution and its distortion. Insurance: Mathematics and Economics, 51(2), 257–264.Google Scholar
Makov, U.E. (1995) Loss robustness via Fisher-weighted squared-error loss function. Insurance: Mathematics and Economics, 16, 1–6.Google Scholar
Makov, U.E., Smith, A.F.M. and Liu, Y.H. (1996) Bayesian methods in actuarial science. The Statistician, 45, 503–515.10.2307/2988548CrossRefGoogle Scholar
Muller, A. and Stoyan, D. (2002) Comparison Methods for Stochastic Models and Risks. New York: John Wiley & Sons.Google Scholar
Ríos, D., Ruggeri, F. and Vidakovic, B. (1995) Some results on posterior regret gamma-minimax estimation. Statistics & Decisions, 13, 315–331.Google Scholar
Schnieper, R. (2004) Robust Bayesian experience rating. ASTIN Bulletin, 34(1), 125–150.CrossRefGoogle Scholar
Shaked, M. and Shanthikumar, J.G. (2007) Stochastic Orders. Springer Series in Statistics. New York: Springer.CrossRefGoogle Scholar
Sordo, M.A. (2008) Characterizations of classes of risk measures by dispersive orders. Insurance: Mathematics and Economics, 42, 1028–1034.Google Scholar
Sordo, M.A., Bello, A. and Suárez-Llorens, A. (2018) Stochastic orders and co-risk measures under positive dependence. Insurance: Mathematics and Economics, 78, 105–113.Google Scholar
Sordo, M.A., Castaño-Martínez, A. and Pigueiras, G. (2016) A family of premium principles based on mixtures of TVaRs. Insurance: Mathematics and Economics, 70, 397–405.Google Scholar
Wang, S. (1996) Premium calculation by transforming the layer premium density. ASTIN Bulletin, 26, 71–92.CrossRefGoogle Scholar
Whitt, W. (1985) Uniform conditional variability ordering of probability distribution. Journal of Applied Probability, 22(3), 619–633.CrossRefGoogle Scholar
Willmot, G.E. (1987) The Poisson-inverse Gaussian distribution as an alternative to the negative binomial. Scandinavian Actuarial Journal, 1987(3-4), 113–127.CrossRefGoogle Scholar
Young, V.R. (1999) Robust Bayesian credibility using semiparametric models. ASTIN Bulletin, 28(1), 187–203.CrossRefGoogle Scholar
Young, V.R. (2004) Premium principles. In Encyclopedia of Actuarial Science (eds. Teugels, J.F. and Sundt, B.), pp. 1322–1331. Chichester: John Wiley & Sons.Google Scholar