Actuarial Modelling of Claim Counts: Risk Classification, Credibility and Bonus-Malus Systems

Actuarial Modelling of Claim Counts: Risk Classification, Credibility and Bonus-Malus Systems

Volumn , Issue , 2007, Pages 1-356

  Denuit, Michel ^a   Maréchal, Xavier ^a   Pitrebois, Sandra ^b   Walhin, Jean François ^c  

^a UNIVERSITÉ CATHOLIQUE DE LOUVAIN   (Belgium)

^b Secura ^*  (Belgium)

^c Fortis   (Belgium)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 84889325128     PISSN: None     EISSN: None     Source Type: Book    
DOI: 10.1002/9780470517420     Document Type: Book

References (231)

1. Albrecht, P. (1983a). Parametric multiple regression risk models: Connections with tariffication, especially in motor insurance. Insurance: Mathematics and Economics 2, 113-117.
2. Albrecht, P. (1983b). Parametric multiple regression risk models: Theory and statistical analysis. Insurance: Mathematics and Economics 2, 49-66.
3. Albrecht, P. (1983c). Parametric multiple regression risk models: Some connections with IBNR. Insurance: Mathematics and Economics 2, 69-73.
4. Albrecht, P. (1984). Laplace transforms, Mellin transforms and mixed Poisson processes. Scandinavian Actuarial Journal, 58-64.
5. Albrecht, P. (1985). An evolutionary credibility model for claim numbers. ASTIN Bulletin 15, 1-17.
6. Ambagaspitya, R.S. (1999). On the distributions of two classes of correlated aggregate claims. Insurance: Mathematics and Economics 24, 255-263.
7. Andrade Silva, J.M., & Centeno, M. (2005). A note on bonus scales. Journal of Risk and Insurance 72, 601-607.
8. Angers, J.-F., Desjardins, D., Dionne, G., & Guertin, F. (2006). Vehicle and fleet random effects in a model of insurance rating for fleets of vehicles. ASTIN Bulletin 36, 25-77.
9. Antonio, K., & Beirlant, J. (2007). Actuarial statistics with generalized linear mixed models. Insurance: Mathematics and Economics 40, 58-76.
10. Arnold, B.C., Castillo, E., & Sarabia, J.M. (1999). Conditional Specification of Statistical Models. Springer, New York.
11. Beirlant, J., Derveaux, V., De Meyer, A.M., Goovaerts, M.J., Labies, E., & Maenhoudt, B. (1991). Statistical risk evaluation applied to (Belgian) car insurance. Insurance: Mathematics and Economics 10, 289-302.
12. Beirlant, J., Goegebeur, Y., Segers, J., & Teugels, J. (2004). Statistics of Extremes: Theory and Applications. Wiley Series in Probability and Statistics. John Wiley & Sons, Ltd.
13. Bermúdez, L., Denuit, M., & Dhaene, J. (2000). Exponential bonus-malus systems integrating a priori risk classification. Journal of Actuarial Practice 9, 84-112.
14. Besson, J.L., & Partrat, C. (1990). Loi de Poisson inverse gaussienne et systèmes de bonus-malus. Proceedings of the Astin Colloquium, Montreux 81, 418-419.
15. Bolancé, C., Guillén, M., & Pinquet, J. (2003). Time-varying credibility for frequency risk models: Estimation and tests for autoregressive specification on the random effect. Insurance: Mathematics and Economics 33, 273-282.
16. Bonsdorff, H. (1992). On the convergence rate of bonus-malus systems. ASTIN Bulletin 22, 217-223.
17. Bonsdorff, H. (2005). On asymptotic properties of bonus-malus systems based on the number and the size of the claims. Scandinavian Actuarial Journal, 309-320.
18. Borgan, O., Hoem, J.M., & Norberg, R. (1981). A nonasymptotic criterion for the evaluation of automobile bonus systems. Scandinavian Actuarial Journal, 165-178.
19. Boskov, M., & Verrall, R.J. (1994). Premium rating by geographical area using spatial models. ASTIN Bulletin 24, 131-143.
20. Boucher, J.-Ph., & Denuit, M. (2006). Fixed versus random effects in Poisson regression models for claim counts: a case study with motor insurance. ASTIN Bulletin 36, 285-301.
21. Boucher, J.-Ph., Denuit, M., & Guillén, M. (2006). Risk classification for claim counts: Zeroinflated mixed Poisson and hurdle models. Working Paper 06-06, Institut des Sciences Actuarielles, Université Catholique de Louvain, Louvain-la-Neuve, Belgium.
22. Brockman, M.J., & Wright, T.S. (1992). Statistical motor rating: Making effective use of your data. Journal of the Institute of Actuaries 119, 457-543.
23. Brouhns, N., Denuit, M., Masuy, B., & Verrall, R.J. (2002). Ratemaking by geographical area in the Boskov and Verrall model: a case study using Belgian car insurance data. actu-L 2, 3-28.
24. Brouhns, N., Guillén, M., Denuit, M., & Pinquet, J. (2003). Bonus-malus scales in segmented tariffs with stochastic migration between segments. Journal of Risk and Insurance 70, 577-599.
25. Buch-Kromann, T. (2006). Estimation of large insurance losses: A case study. Journal of Actuarial Practice 13, 191-211.
26. Buch-Larsen, T., Nielsen, J.P., Guillén, M., & Bolancé, C. (2005). Kernel density estimation for heavy-tailed distributions using the Champernowne transformation. Statistics 39, 503-518.
27. Bühlmann, H. (1967). Experience rating and credibility. ASTIN Bulletin 4, 199-207.
28. Bühlmann, H. (1970). Mathematical Methods in Risk Theory. Springer, New York.
29. Bühlmann, P., & Bühlmann, H. (1999). Selection of credibility regression models. ASTIN Bulletin 29, 245-270.
30. Bühlmann, H., & Gisler, A. (2005). A Course in Credibility Theory and its Applications. Springer, Berlin.
31. Burnham, K.P., & Anderson, D.R. (2002). Model Selection and Multi-Model Inference: A Practical Information-Theoretic Approach. Springer, New York.
32. Butler, P. (1993). Cost-based pricing of individual automobile risk transfer: Carmile exposure unit analysis. Journal of Actuarial Practice 1, 51-84 (with discussion).
33. Cameron, A.C., & Trivedi, P.K. (1998). Regression Analysis of Count Data. Cambridge University Press.
34. Carrière, J. (1993a). Nonparametric tests for mixed Poisson distributions. Insurance: Mathematics and Economics 12, 3-8.
35. Carrière, J. (1993b). A semiparametric estimator of a risk distribution. Insurance: Mathematics and Economics 13, 75-81.
36. Carter, M., & Van Brunt, B. (2000). The Lebesgue-Stieltjes Integral. A Practical Introduction. Springer, New York.
37. Cebrian, A., Denuit, M., & Lambert, Ph. (2003). Generalized Pareto fit to the society of Actuaries' large claims database. North American Actuarial Journal 7, 18-36.
38. Centeno, M., & Andrade Silva, J.M.A. (2001). Bonus systems in an open portfolio. Insurance: Mathematics and Economics 28, 341-350.
39. Centeno, M., & Andrade Silva, J.M.A. (2002). Optimal bonus scales under path-dependent bonus rules. Scandinavian Actuarial Journal, 615-627.
40. Chappell, D., & Norman, J.M. (1989). Optimal, near-optimal and rule-of-thumb claiming rules for a protected bonus vehicle insurance policy. European Journal of Operations Research 41, 151-156.
41. Consul, P.C. (1990). A model for distributions of injuries in auto-accidents. Bulletin of the SWISS Association of Actuaries, 161-168.
42. Cooray, K., & Ananda, M.M.A. (2005). Modeling actuarial data with a composite lognormal-Pareto model. Scandinavian Actuarial Journal, 321-334.
43. Coutts, S. (1984). Motor premium rating. Insurance: Mathematics and Economics 3, 73-96.
44. Cummins, D.J., Dionne, G., McDonnald, J.B., & Pritchett, M.B. (1990). Application of the GB2 family of distributions in modeling insurance loss processes. Insurance: Mathematics and Economics 9, 257-272.
45. Daengdej, J., Lukose, D., & Murison, R. (1999). Using statistical models and case-based reasoning in claims prediction: Experience from a real-world problem. Knowledge-based Systems 12, 239-245.
46. Dalgaard, P. (2002). Introductory Statistics with R. Springer, New York.
47. Dannenburg, D.R., Kaas, R., & Goovaerts, M.J. (1996). Practical Actuarial Credibility Models. Institute of Actuarial Science and Econometrics, University of Amsterdam, Amsterdam, The Netherlands.
48. Dean, C.B., Lawless, J.F., & Willmot, G.E. (1989). A mixed Poisson-inverse-Gaussian regression model. The Canadian Journal of Statistics 17, 171-182.
49. De Boor, C. (1978). A Practical Guide to Splines. Springer, New York.
50. De Leve, G., & Weeda, P.J. (1968). Driving with Markov programming. ASTIN Bulletin 5, 62-86.
51. Dellaert, N.P., Frenk, J.B.G., Kouwenhoven, A., & Van Der Laan, B.S. (1990). Optimal claim behaviour for third party liability insurances, or to claim or not to claim: that is the question. Insurance: Mathematics and Economics 9, 59-76.
52. Dellaert, N.P., Frenk, J.B.G., & van Rijsoort, L.P. (1993). Optimal claim behaviour for vehicle damage insurance. Insurance: Mathematics and Economics 12, 225-244.
53. Dellaert, N.P., Frenk, J.B.G., & Voskol, E. (1991). Optimal claim behaviour for third party liability insurances with perfect information. Insurance: Mathematics and Economics 10, 145-151.
54. Denuit, M. (1997). A new distribution of Poisson-type for the number of claims. ASTIN Bulletin 27, 229-242.
55. Denuit, M. (2002). S-convex extrema, Taylor-type expansions and stochastic approximations. Scandinavian Actuarial Journal, 45-67.
56. Denuit, M., De Vylder, F.E., & Lefèvre, Cl. (1999). Extremal generators and extremal distributions for the continuous s-convex stochastic orderings. Insurance: Mathematics and Economics 24, 201-217.
57. Denuit, M., & Dhaene, J. (2001). Bonus-Malus scales using exponential loss functions. German Actuarial Bulletin 25, 13-27.
58. Denuit, M., Dhaene, J., Goovaerts, M.J., & Kaas, R. (2005). Actuarial Theory for Dependent Risks: Measures, Orders and Models. John Wiley & Sons, Inc., New York.
59. Denuit, M., & Lambert, Ph. (2001). Smoothed NPML estimation of the risk distribution underlying Bonus-Malus systems. Proceedings of the Casualty Actuarial Society 88, 142-174.
60. Denuit, M., & Lang, S. (2004). Nonlife ratemaking with Bayesian GAM's. Insurance: Mathematics and Economics 35, 627-647.
61. Denuit, M., Maréchal, X., Pitrebois, S., & Walhin, J.-F. (2007a). Claiming behaviour in motor insurance with bonus-malus systems. Working Paper, Institut des Sciences Actuarielles, UCL, Louvain-la-Neuve, Belgium.
62. Denuit, M., Maréchal, X., Pitrebois, S., & Walhin, J.-F. (2007b). Actuarial analysis of some special rules in bonus-malus systems. Working Paper, Institut des Sciences Actuarielles, UCL, Louvain-la-Neuve, Belgium.
63. De Pril, N. (1978). The efficiency of a Bonus-Malus system. ASTIN Bulletin 10, 59-72.
64. De Pril, N. (1979). Optimal claim decisions for a Bonus-Malus system: A continuous approach. ASTIN Bulletin 10, 215-222.
65. De Pril, N., & Goovaerts, M.J. (1983). Bounds for the optimal critical claim size of a bonus system. Insurance: Mathematics and Economics 2, 27-32.
66. Der, G., & Everitt, B. (2002). A Handbook of Statistical Analyses using SAS®. Chapmann & Hall/CRC, Boca Raton.
67. Desjardins, D., Dionne, G., & Pinquet, J. (2001). Experience rating schemes for fleets of vehicles. ASTIN Bulletin 31, 81-105.
68. De Vylder, F.E. (1985). Non-linear regression in credibility theory. Insurance: Mathematics and Economics 4, 163-172.
69. De Vylder, F.E. (1996). Advanced Risk Theory. A Self-Contained Introduction. Editions de l'Université Libre de Bruxelles-Swiss Association of Actuaries, Bruxelles.
70. De Wit, G.W., & Van Eeghen, J. (1984). Rate making and society's sense of fairness. ASTIN Bulletin 14, 151-163.
71. Dhaene, J., & Vandebroek, M. (1995). Recursions for the individual model. Insurance: Mathematics and Economics 16, 31-38.
72. Dharmadhikari, S.W., & Joag-Dev, K. (1988). Unimodality, Convexity and Applications. Academic Press, New York.
73. Dickson, D.C.M., & Waters, H.R. (1999). Multi-period aggregate loss distributions for a life portfolio. ASTIN Bulletin 29, 295-309.
74. Dimakos, X.K., & Rattalma, A.F. (2002). Bayesian premium rating with latent structure. Scandinavian Actuarial Journal, 162-184.
75. Dionne, G., Artis, M., & Guillén, M. (1996). Count data models for a credit scoring system. Journal of Empirical Finance 3, 303-325.
76. Dionne, G., & Vanasse, C. (1989). A generalization of actuarial automobile insurance rating models: the Negative Binomial distribution with a regression component. ASTIN Bulletin 19, 199-212.
77. Dionne, G., & Vanasse, C. (1992). Automobile insurance ratemaking in the presence of asymmetrical information. Journal of Applied Econometrics 7, 149-165.
78. Dixon, M., Kelsey, R., & Verrall, R. (2000). Postcode insurance rating: spatial modelling and performance evaluation. Paper presented at the 4th IME Congress, Barcelona.
79. Dufresne, F. (1988). Distribution stationnaire d'un système bonus-malus et probabilité de ruine. ASTIN Bulletin 18, 31-46.
80. Dufresne, F. (1995). The efficiency of the Swiss Bonus-Malus system. Bulletin of the Swiss Association of Actuaries, 29-41.
81. Elvers, E. (1991). A note on the Generalized Poisson distribution. ASTIN Bulletin 21, 167.
82. Fahrmeir, L., Lang, S., & Spies, F. (2003). Generalized geoadditive models for insurance claims data. German Actuarial Bulletin 26, 7-23.
83. Fahrmeir, L., & Tutz, G. (2001). Multivariate Statistical Modelling Based on Generalized Linear Models. Springer, New York.
84. Faraway, J.J. (2006). Extending the Linear Model with R. Chapman & Hall/CRC, Boca Raton.
85. Feller, W. (1971). An Introduction to Probability Theory and its Applications (Vol. II). John Wiley & Sons, Inc., New York.
86. Ferreira, J. (1977). Identifying equitable insurance premiums for risk classes: an alternative to the classical approach. Lecture presented at the 23rd international meeting of the Institute of Management Sciences, Athens, Greece.
87. Franklin, C.H. (2005). Maximum likelihood estimation. In: Encyclopedia of Social Measurement, Vol. 2, pp. 653-664. John Wiley & Sons, Inc., New York.
88. Frees, E.W. (2003). Multivariate credibility for aggregate loss models. North American Actuarial Journal 7, 13-27.
89. Frees, E.W., & Wang, P. (2005). Credibility using copulas. North American Actuarial Journal 9, 31-48.
90. Frees, E.W., & Wang, P. (2006). Copula credibility for aggregate loss models. Insurance: Mathematics and Economics 38, 360-373.
91. Frees, E.W., Young, V.R., & Luo, Y. (1999). A longitudinal data analysis interpretation of credibility models. Insurance: Mathematics and Economics 24, 229-247.
92. French, E., & Jones, J.B. (2004). On the distribution and dynamics of health care costs. Journal of Applied Econometrics 19, 705-722.
93. Gerber, H.U., & Jones, D. (1975). Credibility formulas of the updating type. Transactions of the Society of Actuaries 27, 31-52.
94. Gertensgarbe, F.W., & Werner, P.C. (1989). A method for the statistical definition of extreme-value regions and their application to meteorological time series. Zeitschrift fur Meteorologie 39, 224-226.
95. Gilde, V., & Sundt, B. (1989). On bonus systems with credibility scales. Scandinavian Actuarial Journal, 13-22.
96. Golden, R.M. (2003). Discrepancy risk model selection test theory for comparing possibly misspecified or nonnested models. Psychometrika 68, 229-249.
97. Goovaerts, M.J., & Hoogstad, W. (1987). Credibility Theory. Survey in Actuarial Studies, Nationale Nederlanden N.V., Rotterdam.
98. Gossiaux, A.-M., & Lemaire, J. (1981). Méthodes d'ajustement de distributions de sinistres. Bulletin of the Swiss Association of Actuaries, 87-95.
99. Grandell, J. (1997). Mixed Poisson Processes. Chapman & Hall, New York.
100. Greenwood, M., & Yule, G.U. (1920). An inquiry into the nature of frequency distributions representative of multiple happenings with particular reference to the occurrence of multiple attacks of disease or of repeated accidents. Journal of the Royal Statistical Society, Series A 83, 255-279.
101. Grenander, U. (1957a). On the heterogeneity in non-life insurance. Scandinavian Actuarial Journal, 153-179.
102. Grenander, U. (1957b). Some remarks on bonus systems in automobile insurance. Scandinavian Actuarial Journal, 180-197.
103. Hachemeister, C.A. (1975). Credibility for regression models with application to trend. In: P.M. Kahn, Editor, Credibility: Theory and Applications, Academic Press, New York, pp. 129-163.
104. Haehling von Lanzenauer, C. (1974). Optimal claim decisions by policyholders in automobile insurance with merit rating structures. Operations Research 22, 979-990.
105. Hastings, N.A.J. (1976). Optimal claiming on vehicle insurance. Operational Research Quarterly 27, 908-913.
106. Henriet, D., & Rochet, J.C. (1986). La logique des systèmes bonus-malus en assurance automobile: une approche théorique. Annales d'Économie et de Statistique 1, 133-152.
107. Heras, A., Gil, J.A., Garcia-Pineda, P., & Vilar, J.C. (2004). An application of linear programming to bonus-malus system design. ASTIN Bulletin 34, 435-456.
108. Heras, A., Vilar, J.L., & Gil, J.A. (2002). Asymptotic fairness of bonus-malus systems and optimal scales of premiums. The Geneva Papers on Risk and Insurance-Theory 27, 61-82.
109. Herzog, T.N. (1994). Introduction to Credibility Theory. ACTEX Publications.
110. Hey, J.D. (1985). No claims bonus. The Geneva Papers on Risk and Insurance 10, 209-228.
111. Hinde, J. (1982). Compound Poisson regression models. In: R. Gilchrist, Editor, GLIM 82: Proceedings of the International Conference on Generalised Linear Models, Springer, New York.
112. Holtan, J. (1994). Bonus made easy. ASTIN Bulletin 24, 61-74.
113. Holtan, J. (2001). Optimal loss financing under bonus-malus contracts. ASTIN Bulletin 31, 161-173.
114. Hsiao, C. (2003). Analysis of Panel Data. Cambridge University Press, Cambridge.
115. Huang, X., Song, L., & Liang, Y. (2003). Semiparametric credibility ratemaking using a piecewise linear prior. Insurance: Mathematics and Economics 33, 585-593.
116. Hürlimann, W. (1990) On maximum likelihood estimation for count data models. Insurance: Mathematics and Economics 9, 39-49.
117. Islam, M.N., & Consul, P.C. (1992). A probabilistic model for automobile claims. Bulletin of the Swiss Association of Actuaries 85-93.
118. Jee, B. (1989). A comparative analysis of alternative pure premium models in the automobile risk classification system. Journal of Risk and Insurance 56, 434-459.
119. Jewell, W.S. (1975). Model variations in credibility theory. In: P.M. Kahn, Editor, Credibility, Academic Press, New York, pp. 199-244.
120. Johnson, N.L., Kotz, S., & Kemp, A.W. (1992). Univariate Discrete Distributions, John Wiley & Sons, Inc., New York.
121. Jorgensen, B. (1987). Exponential dispersion models. Journal of the Royal Statistical Society, Series B 49, 127-162.
122. Jorgensen, B. (1997). The Theory of Regression Models. Chapman & Hall, London.
123. Jorgensen, B., & Paes de Souza, M.C. (1994). Fitting Tweedie's compound Poisson model to insurance claims data. Scandinavian Actuarial Journal, 69-93.
124. Karlis, D. (2005). EM algorithm for mixed Poisson and other discrete distributions. ASTIN Bulletin 35, 3-24.
125. Kelle, M. (2000). Modélisation du système de bonus malus français. Bulletin Français d'Actuariat 4, 45-64.
126. Kendall, M., & Stuart, A. (1977). Advanced Theory of Statistics, Vol. I. Griffin, London.
127. Kestemont, R.-M., & Paris, J. (1985). Sur l'ajustement du nombre de sinistres. Bulletin of the Swiss Association of Actuaries, 157-163.
128. Klugman, S. (1992). Bayesian Statistics in Actuarial Science. Kluwer, Boston.
129. Klugman, S., Panjer, H., & Willmot, G. (2004). Loss Models: From Data to Decisions. John Wiley & Sons, Inc., New York.
130. Kolderman, J., & Volgenant, A. (1985). Optimal claiming in an automobile insurance system with bonus-malus structure. Journal of the Operational Research Society 36, 239-247.
131. Lambert, D. (1992). Zero-inflated Poisson regression, with an application to defects in manufacturing. Technometrics 34, 1-14.
132. Lee, Y., & Nelder, J.A. (1996). Hierarchical generalized linear models. Journal of the Royal Statistical Society, Series B 58, 619-678.
133. Lefèvre, Cl., & Picard, Ph. (1996). On the first crossing of a Poisson process in a lower boundary. In: C.C. Heyde, Yu V. Prohorov, R. Pyke and S.T. Rachev Editors, Athens Conference on Applied Probability and Time Series, Vol. I, Applied Probability. Lecture Notes in Statistics, Springer, Berlin, 114, pp. 159-175.
134. Lemaire, J. (1976). Driver versus company: Optimal behaviour of the policyholder. Scandinavian Actuarial Journal, 209-219.
135. Lemaire, J. (1977). La soif du bonus. ASTIN Bulletin 9, 181-190.
136. Lemaire, J. (1979). How to define a Bonus-Malus system with an exponential utility function. ASTIN Bulletin 10, 274-282.
137. Lemaire, J. (1995). Bonus-Malus Systems in Automobile Insurance. Kluwer Academic Publisher, Boston.
138. Lemaire, J., & Vandermeulen, E. (1983). Une propriété du principe de l'espérance mathématique. Bulletin Trimestriel de l'Institut des Actuaires Français, 5-14.
139. Lemaire, J., & Zi, H. (1994a). High deductibles instead of Bonus-Malus. Can it work? ASTIN Bulletin 24, 75-88.
140. Lemaire, J., & Zi, H. (1994b). A comparative analysis of 30 Bonus-Malus systems. ASTIN Bulletin 24, 287-309.
141. Liang, K.Y., & Zeger, S.L. (1986). Longitudinal data analysis using generalized linear models. Biometrika 73, 13-22.
142. Lindsay, B. (1989a). On the determinants of moment matrices. Annals of Statistics 17, 711-721.
143. Lindsay, B. (1989b). Moment matrices: applications in mixtures. Annals of Statistics 17, 722-740.
144. Lindsay, B. (1995). Mixture Models: Theory, Geometry and Applications. Institute of Mathematical Statistics and the American Statistical Association.
145. Lo, C.H., Fung, W.K., & Zhu, Z.Y. (2006). Generalized estimating equations for variance and covariance parameters in regression credibility models. Insurance: Mathematics and Economics 39, 99-113.
146. Loimaranta, K. (1972). Some asymptotic properties of bonus systems. ASTIN Bulletin 6, 223-245.
147. Lord, D. (2006). Modeling motor vehicle crashes using Poisson-gamma models: Examining the effects of low sample mean values and small sample size on the estimation of the fixed dispersion parameter. Accident Analysis and Prevention 38, 751-766.
148. Luo, Y., Young, V.R., & Frees, E.W. (2004). Credibility ratemaking using collateral information. Scandinavian Actuarial Journal, 448-461.
149. Makov, U.E. (2002). Principal applications of Bayesian methods in actuarial science: A perspective. North American Actuarial Journal 5, 53-73 (with discussion).
150. Makov, U.E., Smith, A.F.M., & Liu, Y.H. (1996). Bayesian methods in actuarial science. The Statistician 45, 503-517.
151. Martin-Lof, A. (1973). A method for finding the optimal decision rule for a policy holder of an insurance with a bonus system. Scandinavian Actuarial Journal, 23-29.
152. Mc Cullagh, P., & Nelder, J.A. (1989). Generalized Linear Models. Chapman & Hall, New York.
153. Morillo, I., & Bermudez, L. (2003). Bonus-malus systems using an exponential loss function with an Inverse Gaussian distribution. Insurance: Mathematics and Economics 33, 49-57.
154. Mowbray, A.H. (1914). How extensive a payroll exposure is necessary to give a dependable pure premium. Proceedings of the Casualty Actuarial Society 1, 24-30.
155. Nelder, J.A., & Verral, R.J. (1997). Credibility theory and generalized linear models. ASTIN Bulletin 27, 71-82.
156. Nelder, J.A., & Wedderburn, R.W.M. (1972). Generalized linear models. Journal of the Royal Statistical Society, Series A 135, 370-384.
157. Neuhaus, W. (1988). A bonus-malus system in automobile insurance. Insurance: Mathematics and Economics 7, 103-112.
158. Norberg, R. (1975). Credibility premium plans which make allowance for bonus hunger. Scandinavian Actuarial Journal, 73-86.
159. Norberg, R. (1976). A credibility theory for automobile bonus system. Scandinavian Actuarial Journal, 92-107.
160. Norberg, R. (1980). Empirical Bayes credibility. Scandinavian Actuarial Journal, 177-194.
161. Norberg, R. (1986). Hierarchical credibility: Analysis of a random effect linear model with nested classification. Scandinavian Actuarial Journal, 204-222.
162. Norberg, R. (2004). Credibility Theory. In: J. Teugels and B. Sundt Editors, Encyclopedia of Actuarial Science, John Wiley & Sons, Ltd, pp. 398-406.
163. Norman, J.M., & Shearn, D.C.S. (1980). Optimal claiming on vehicle insurance revisited. Journal of the Operational Research Society 31, 181-186.
164. Panjer, H.H. (1981). Recursive evaluation of a family of compound distributions. ASTIN Bulletin 12, 22-26.
165. Panjer, H.H. (1987). Models of claim frequency. In: I.B. Mac Neill and G.J. Umphrey Editors, Advances in the Statistical Sciences, Vol. VI, Actuarial Sciences, The University of Western Ontario Series in Philosophy of Science, Reidel, Dordrecht 39, pp. 115-122.
166. Philipson, C. (1960). The Swedish system of bonus. ASTIN Bulletin 1, 134-141.
167. Picard, P. (1976). Généralisation de l'étude sur la survenance des sinistres en assurance automobile. Bulletin Trimestriel de l'Institut des Actuaires Français, 204-267.
168. Pinquet, J. (1997). Allowance for cost of claims in Bonus-Malus systems. ASTIN Bulletin 27, 33-57.
169. Pinquet, J. (1998). Designing optimal bonus-malus systems from different types of claims, ASTIN Bulletin, 205-220.
170. Pinquet, J. (2000). Experience rating through heterogeneous models. In: G. Dionne, Editor, Handbook of Insurance, Kluwer Academic Publishers.
171. Pinquet, J., Guillén, M., & Bolancé, C. (2001). Allowance for the age of claims in bonus-malus systems. ASTIN Bulletin 31, 337-348.
172. Pitrebois, S., Denuit, M., & Walhin, J.-F. (2003a). Fitting the Belgian Bonus-Malus system. Belgian Actuarial Bulletin 3, 58-62.
173. Pitrebois, S., Denuit, M., & Walhin, J.-F. (2003b). Setting a bonus-malus scale in the presence of other rating factors: Taylor's work revisited. ASTIN Bulletin 33, 419-436.
174. Pitrebois, S., Denuit, M., & Walhin, J.-F. (2004). Bonus-malus scales in segmented tariffs: Gilde & Sundt's work revisited. Australian Actuarial Journal 10, 107-125.
175. Pitrebois, S., Denuit, M., & Walhin, J.-F. (2005). Bonus-malus systems with varying deductibles. ASTIN Bulletin 35, 261-274.
176. Pitrebois, S., Denuit, M., & Walhin, J.-F. (2006a). Multi-event bonus-malus scales. Journal of Risk and Insurance 73, 517-528.
177. Pitrebois, S., Denuit, M., & Walhin, J.-F. (2006b). An actuarial analysis of the French bonus-malus system. Scandinavian Actuarial Journal, 247-264.
178. Pitrebois, S., Walhin, J.-F., & Denuit, M. (2006c). How to transfer policyholders from one bonusmalus scale to the other? German Actuarial Bulletin 27, 607-618.
179. Pohlmeier, W., & Ulrich, V. (1995). An econometric model of the two-part decision making process in the demand for health care. Journal of Human Resources 30, 339-361.
180. Purcaru, O., & Denuit, M. (2003). Dependence in dynamic claim frequency credibility models. ASTIN Bulletin 33, 23-40.
181. Qian, W. (2000). An application of nonparametric regression estimation in credibility theory. Insurance: Mathematics and Economics 27, 169-176.
182. Quigley, J., Bedford, T., & Walls, L. (2006). Estimating rate of occurrence of rare events with empirical bayes: A railway application. Reliability Engineering and System Safety, in press.
183. Raftery, A.E. (1995). Bayesian model selection in social research. In: P. Marsden, Editor, Sociological Methodology, The American Sociological Association, Washington, pp. 111-163.
184. Rejesus, R.M., Coble, K.H., Knight, T.O., & Jin, Y. (2006). Developing experience-based premium rate discounts in crop insurance. American Journal of Agricultural Economics 88, 409-419.
185. Retzlaff-Roberts, D., & Puelz, R. (1996). Classification in automobile insurance using a DEA and discriminant analysis hybrid. Journal of Productivity Analysis 7, 417-427.
186. Roberts, A.W., & Varberg, D.E. (1973). Convex Functions. Academic Press, New York.
187. Rolski, T., Schmidli, H., Schmidt, V., & Teugels, J. (1999). Stochastic Processes for Insurance and Finance. John Wiley & Sons, Inc., New York.
188. Ruohonen, M. (1987). On a model for the claim number process. ASTIN Bulletin 18, 57-68.
189. Santos Silva, J.M.C., & Windmeijer, F. (2001). Two-part multiple spell models for health care demand. Journal of Econometrics 104, 67-89.
190. Sarabia, J.M., Gomez-Deniz, E., & Vazquez-Polo, F.J. (2004). On the use of conditional specification models in claim count distributions: An application to bonus-malus systems. ASTIN Bulletin 34, 85-98.
191. Shaked, M. (1980). On mixtures from exponential families. Journal of the Royal Statistical Society, Series B 42, 192-198.
192. Sharif, A.H., & Panjer, H.H. (1993). A probabilistic model for automobile claims: a comment on the article by M.N. Islam and P.C. Consul. Bulletin of the Swiss Association of Actuaries, 279-282.
193. Shengwang, M., Wei, Y., & Whitmore, G.A. (1999). Accounting for individual over-dispersion in a Bonus-Malus automobile insurance system. ASTIN Bulletin 29, 327-337.
194. Simar, L. (1976). Maximum likelihood estimation of a compound Poisson process. The Annals of Statistics 4, 1200-1209.
195. Smyth, G.J., & Jorgensen, B. (2002). Fitting Tweedie's compound Poisson model to insurance claims data: Dispersion modelling. ASTIN Bulletin 32, 143-157.
196. Subramanian, K. (1998). Bonus-malus systems in a competitive environment. North American Actuarial Journal 2, 38-44.
197. Sundt, B. (1981). Recursive credibility estimation. Scandinavian Actuarial Journal, 3-21.
198. Sundt, B. (1983). Parameter estimation in some credibility models. Scandinavian Actuarial Journal, 239-255.
199. Sundt, B. (1988). Credibility estimators with geometric weights. Insurance: Mathematics and Economics 7, 113-122.
200. Sundt, B. (1999). On multivariate Panjer recursions. ASTIN Bulletin 29, 29-45.
201. Taylor, G.C. (1989). Use of spline functions for premium rating by geographic area. ASTIN bulletin 19, 91-122.
202. Taylor, G.C. (1996). Geographic premium rating by Whittaker spatial smoothing. ASTIN Bulletin 31, 147-160.
203. Taylor, G. (1997). Setting a bonus-malus scale in the presence of other rating factors. ASTIN Bulletin 27, 319-327.
204. ter Berg, P. (1980a). Two pragmatic approaches to loglinear claim cost analysis. ASTIN Bulletin 11, 77-90.
205. ter Berg, P. (1980b). On the loglinear Poisson and gamma model. ASTIN Bulletin 11, 35-40.
206. ter Berg, P. (1996). A loglinear lagrangian Poisson model. ASTIN Bulletin 26, 123-129.
207. Titterington, D.M., Smith, A.F.M., & Makov, U.E. (1985). Statistical Analysis of Finite Mixture Distributions. John Wiley & Sons Ltd, Chichester.
208. Tremblay, L. (1992). Using the Poisson-inverse gaussian distribution in bonus-malus systems. ASTIN Bulletin 22, 97-106.
209. Tucker, H.G. (1963). An estimate of the compounding distribution of a compound Poisson distribution. Theory of Probability and Its Applications 8, 195-200.
210. Tweedie, M.C.K. (1984). An index which distinguishes between some important exponential families. In: J.K. Ghosh and J. Roy, Editors, Statistics: Applications and New Directions, Proceedings of the Indian Statistical Institute Golden Jubilee International Conference, pp. 579-604.
211. Vandebroek, M. (1993). Bonus-malus system or partial coverage to oppose moral hazard problems? Insurance: Mathematics and Economics 13, 1-5.
212. Viswanathan, K.S., & Lemaire, J. (2005). Bonus-malus systems in a deregulated environment: Forecasting market shares using diffusion models. ASTIN Bulletin 35, 299-319.
213. Vuong, Q.H. (1989). Likelihood ratio tests for model selection and non-nested hypotheses. Econometrica 57, 307-333.
214. Walhin, J.-F. (2001). Some comments on the individual risk model and multivariate extension. German Actuarial Bulletin 25, 257-270.
215. Walhin, J.-F., & Paris, J. (1999). Using mixed Poisson distributions in connection with Bonus-Malus systems. ASTIN Bulletin 29, 81-99.
216. Walhin, J.-F., & Paris, J. (2000). The true claim amount and frequency distributions within a bonusmalus system. ASTIN Bulletin 30, 391-403.
217. Walhin, J.-F., & Paris, J. (2001). The practical replacement of a bonus-malus system. ASTIN Bulletin 31, 317-335.
218. Wedel, M., Böckenholtb, U., & Kamakurac, W.A. (2003). Factor models for multivariate count data. Journal of Multivariate Analysis 87, 356-369.
219. Weisberg, H.I., Tomberlin, T.J., & Chatterjee, S. (1984). Predicting insurance losses under crossclassification: A comparison of alternative approaches. Journal of Business & Economic Statistics 2, 170-178.
220. Whitney, W. (1918). The theory of experience rating. Proceedings of the Casualty Actuarial Society 4, 274-292.
221. Williams, G.J., & Huang, Z. (1996). KDD for insurance risk assessment: A case study. CSIRO Mathematical and Information Sciences.
222. Willmot, G.E. (1987). The Poisson-inverse gaussian distribution as an alternative to the negative binomial. Scandinavian Actuarial Journal, 113-127.
223. Winkelmann, R. (2003). Econometric Analysis of Count Data. Springer, New York.
224. Yeo, A.C., Smith, K.A., Willis, R.J., & Brooks, M. (2001). Clustering technique for risk classification and prediction of claim costs in the automobile insurance industry. International Journal of Intelligent Systems in Accounting, Finance and Management 10, 39-50.
225. Yeo, K.L., & Valdez, E.A. (2006). Claim dependence with common effects in credibility models. Insurance: Mathematics and Economics 38, 609-629.
226. Yip, K.C.H., & Yau, K.K.W. (2005). On modeling claim frequency data in general insurance with extra zeros. Insurance: Mathematics and Economics 36, 153-163.
227. Young, V.R. (1997). Credibility using semiparametric models. ASTIN Bulletin 27, 273-285.
228. Young, V.R. (1998a). Credibility using a loss function from spline theory: parametric models with a one-dimensional sufficient statistic. North American Actuarial Journal 2, 101-117 (with discussion).
229. Young, V.R. (1998b). Robust Bayesian credibility using semiparametric models. ASTIN Bulletin 28, 187-203.
230. Young, V.R. (2000). Credibility using semiparametric models and a loss function with a constancy penalty. Insurance: Mathematics and Economics 26, 151-156.
231. Young, V.R., & De Vylder, F.E. (2000). Credibility in favor of unlucky insureds. North American Actuarial Journal 4, 107-113.