Abstract
The conditional specification technique introduced by Arnold et al. (Conditional specification of statistical models. Springer series in statistics. Springer, New York, 1999) was used in Sarabia et al. (Astin Bull 34(1):85–98, 2004) to obtain bonus-malus premiums. The Poisson distribution for which the parameter is a function of the classical structure parameter was used and a new class of prior distributions appeared in a natural way. This model contains, as a particular case, the classical compound Poisson model. In the present paper, the Bayesian robustness of this new model is examined and found to be much more robust than in the classical model in Gómez et al. (Insur Math Econ 31:105–113, 2002). For the present study, the moment conditions on the prior distribution are required. Examples, with real data, are given to illustrate our ideas under the net and exponential premium principles.
Similar content being viewed by others
References
Arnold BC, Castillo E, Sarabia JM (1999) Conditional specification of statistical models. Springer series in statistics. Springer, New York
Betrò B, Ruggeri F, Mȩczarski M (1994) Robust Bayesian analysis under generalized moments conditions. J Stat Plan Inf 41:257–266
Dey DK, Micheas AC (2000) Ranges of posterior expected losses and ɛ-robust actions. In: Ríos Insua D, Ruggeri F (eds) Robust Bayesian analysis. Lecture Notes in Statistics. Springer, New York
Frangos N, Vrontos S (2001) Design of optimal bonus-malus systems with a frequency and a severity component on an individual basis in automobile insurance. Astin Bull 31(1):1–22
Gómez E, Pérez J, Hernández A, Vázquez FJ (2002) Measuring sensitivity in a bonus-malus system. Ins Math Econ 31:105–113
Gómez E, Vázquez FJ (2005) Modelling uncertainty in insurance bonus-malus premiums by using a Bayesian robustness approach. J Appl Stat 32(7):771–784
Heilmann W (1989) Decision theoretic foundations of credibility theory. Ins Math Econ 8:77–95
Jain GC, Consul PC (1971) A generalized negative binomial distribution. SIAM J Appl Math 21(4):501–513
Klugman S, Panjer H, Willmot G (1998) Loss models. From data to decisions. Wiley, New York
Lemaire J (1979) How to define a Bonus-Malus system with an exponential utility function. Astin Bull 10:274–282
Moreno E, Bertolino F, Racugno W (2003) Bayesian inference under partial prior information. Scand J Stat 30:565–580
O’Hagan A, Forster J (2004) Bayesian inference. Kendall’s Advanced Theory and Statistics. Arnold Publishers, London
Pitrebois S, Denuit M, Walhin JF (2006) An actuarial analysis of the French bonus-malus system. Scand Actuarial J 5:247–264
Ríos D, Ruggeri F (2000) Robust Bayesian analysis. Lecture Notes in Statistics. Springer, New York
Sarabia J, Gómez E, Vázquez F (2004) On the use of conditional specification models in claim count distributions: an application to Bonus-Malus systems. Astin Bull 34(1):85–98
Winkler G (2000) Moment sets of bell-shaped distributions: extreme points, extremal decomposition and Chebisheff inequalities. Math Nachr 215:161–184
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Déniz, E.G., Sarabia, J.M. & Vázquez Polo, F.J. Robust Bayesian bonus-malus premiums under the conditional specification model. Stat Papers 50, 465–480 (2009). https://doi.org/10.1007/s00362-007-0085-0
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00362-007-0085-0