Skip to main content
Disciplines
Automotive Business IT + Informatics Construction + Real Estate Electrical Engineering + Electronics Energy + Sustainability Insurance + Risk Finance + Banking Management + Leadership Marketing + Sales Mechanical Engineering + Materials
Events
DE
EN
Books
Journals
Topic Page
Marketing
Start single access now
Access for companies
EXTENDED SEARCH
Log in
Top
European Actuarial Journal
Published in: European Actuarial Journal 1/2019

17-11-2018 | Original Research Paper

The negative binomial-inverse Gaussian regression model with an application to insurance ratemaking

Authors: G. Tzougas, W. L. Hoon, J. M. Lim

Published in: European Actuarial Journal | Issue 1/2019

Log in

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Abstract

This paper presents the Negative Binomial-Inverse Gaussian regression model for approximating the number of claims as an alternative to mixed Poisson regression models that have been widely used in various disciplines including actuarial applications. The Negative Binomial-Inverse Gaussian regression model can be considered as a plausible model for highly dispersed claim count data and this is the first time that it is used in a statistical or actuarial context. The main achievement is that we propose a quite simple Expectation-Maximization type algorithm for maximum likelihood estimation of the model. Finally, a real data application using motor insurance data is examined and both the a priori and a posteriori, or Bonus-Malus, premium rates resulting from the Negative Binomial-Inverse Gaussian model are calculated via the net premium principle and compared to those determined by the Negative Binomial Type I and the Poisson-Inverse Gaussian regression models that have been traditionally used for a priori and a posteriori ratemaking.
previous article Bivariate regular variation among randomly weighted sums in general insurance
next article Discussion on “The negative binomial-inverse Gaussian regression model with an application to insurance ratemaking” (by Tzougas et al.)

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

inform now
Footnotes
1
Note that the NBIG distribution given in Eq. (5) is different from the one used in Gómez-Deniz et al. [24], who considered the case without covariate information.
 
2
Note that pseudo-values will be used in lieu of the original values to estimate the parameters of interest.
 
3
Note that the location and scale parameters of the NBI and PIG models are denoted by \(\mu\) and \(\sigma\) respectively
 
4
Note that all the explanatory variables and the parameters of the models are statistically significant at a 5% threshold.
 
5
We also used three fourths of the data set to estimate the parameters of the models and the remaining one fourth was used to test the out-of-sample prediction accuracy of the models. As expected, our findings were consistent with those provided by the AIC criterion. For more details, refer to Stone [46] who showed that AIC and leave-one out cross validation are asymptotically equivalent.
 
Literature
1.
2.
3.
Bozdogan H (1987) Model selection and Akaike’s Information Criterion (AIC). The general theory and its analytical extensions. Psychometrika 52(3):345–370MathSciNetCrossRefMATH
4.
Bohning D (1999) Computer assisted analysis of mixtures and applications in meta-analysis, disease mapping and others. CRC Press, New YorkMATH
5.
Booth JG, Hobert JP (1999) Maximizing generalized linear mixed model likelihoods with an automated Monte Carlo EM algorithm. J R Stat Soc Ser B (Stat Methodol) 61(1):265–285CrossRefMATH
6.
Booth JG, Hobert JP, Jank W (2001) A survey of Monte Carlo algorithms for maximizing the likelihood of a two-stage hierarchical model. Stat Model 1(4):333–349CrossRefMATH
7.
Boucher JP, Denuit M, Guillen M (2007) Risk classification for claim counts: a comparative analysis of various Zero-Inflated mixed Poisson and Hurdle models. N Am Actuar J 11(4):110–131MathSciNetCrossRef
8.
Boucher JP, Denuit M, Guillen M (2008) Models of insurance claim counts with time dependence based on generalisation of Poisson and negative binomial distributions. Variance 2(1):135–162
9.
Brouhns N, Guillen M, Denuit M, Pinquet J (2003) Bonus-malus scales in segmented tariffs with stochastic migration between segments. J Risk Insur 70:577–599CrossRef
10.
11.
12.
De Jong P, Heller GZ (2008) Generalized linear models for insurance data, vol 10. Cambridge University Press, CambridgeCrossRefMATH
13.
Dempster AP, Laird NM, Rubin DB (1977) Maximum likelihood from incomplete data via the EM algorithm. J R Stat Soc Ser B (Methodol) 39(1):1–38MathSciNetMATH
14.
Denuit M, Lang S (2004) Nonlife ratemaking with Bayesian GAM’s. Insur Math Econ 35:627–47CrossRef
15.
Denuit M, Marechal X, Pitrebois S, Walhin JF (2007) Actuarial modelling of claim counts: risk classification, credibility and bonus-malus systems. Wiley, HobokenCrossRefMATH
16.
Dionne G, Vanasse C (1989) A generalization of actuarial automobile insurance rating models: the negative binomial distribution with a regression component. ASTIN Bull 19:199–212CrossRef
17.
Dionne G, Vanasse C (1992) Automobile insurance ratemaking in the presence of asymmetrical information. J Appl Econ 7:149–165CrossRef
18.
Dunn PK, Smyth GK (1996) Randomized quantile residuals. Comput Graph Stat 5:236–245
19.
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):1–22MathSciNetCrossRefMATH
20.
Frees EW (2010) Regression modeling with actuarial and financial applications. Cambridge University Press, CambridgeMATH
21.
Geedipally SR, Lord D, Dhavala SS (2012) The negative binomial-Lindley generalized linear model: characteristics and application using crash data. Accid Anal Prev 45:258–265CrossRef
22.
Gençtürk Y, Yiğiter A (2016) Modelling claim number using a new mixture model: negative binomial gamma distribution. J Stat Comput Simul 86(10):1829–1839MathSciNetCrossRef
23.
Gómez-Déniz E, Vázquez F (2003) Robustness in Bayesian model for Bonus-Malus systems. In: Intelligent and Other Computation Techniques in Insurance. Theory and Applications. World Scientific
24.
Gómez-Déniz E, Sarabia JM, Calderín-Ojeda E (2008) Univariate and multivariate versions of the negative binomial-inverse Gaussian distributions with applications. Insur Math Econ 42(1):39–49MathSciNetCrossRefMATH
25.
Gómez-Déniz E, Hernández-Bastida A, Fernández-Sánchez MP (2014) Computing credibility bonus-malus premiums using the aggregate claims distribution. Hacettepe J Math Stat 43(6):1047–1061MathSciNetMATH
26.
Gómez-Déniz E, Calderín-Ojeda E (2016) The mixture Poisson exponential-inverse Gaussian regression model: an application in health services. Metodoloski Zvezki 13(2):71
27.
Gómez-Déniz E, Calderín-Ojeda E (2017) An alternative representation of the negative binomial-Lindley distribution. New results and applications. arXiv preprint arXiv:​1703.​04812
28.
Haberman S, Renshaw AE (1996) Generalized linear models and actuarial science. The Statistician, 407-436
29.
30.
Jørgensen B (1982) Statistical Properties of the Generalized Inverse Gaussian Distribution. In: Lecture Notes in Statistics, vol 9. Springer, New York
31.
Kaas R, Goovaerts M, Dhaene J, Denuit M (2008) Modern actuarial risk theory: using R, 2nd edn. Springer, Berlin, HeidelbergCrossRefMATH
32.
Karlis D (2001) A general EM approach for maximum likelihood estimation in mixed Poisson regression models. Stat Model 1(4):305–318CrossRefMATH
33.
34.
Klugman S, Panjer H, Willmot G (1998) Loss models. From data to decisions. Wiley, New YorkMATH
35.
Lambert D (1992) Zero-inflated Poisson regression, with an application to defects in manufacturing. Technometrics 34(1):1–14CrossRefMATH
36.
Lemaire J (1995) Bonus-Malus systems in automobile insurance. Kluwer Academic Publishers, DordrechtCrossRef
37.
Lemaire J, Park SC, Wang KC (2015) The impact of covariates on a Bonus-Malus system: an application of Taylor’s model. Eur Actuar J 5(1):1–10MathSciNetCrossRef
38.
39.
Louis TA (1982) Finding the observed information matrix when using the EM algorithm. J R Stat Soc B 44:226–233MathSciNetMATH
40.
McLachlan G, Krishnan T (2007) The EM algorithm and extensions, vol 382. Wiley, HobokenMATH
41.
Picech L (1994) The merit-rating factor in a multiplicating rate-making model. ASTIN Colloquium, Cannes
42.
Pinquet J (1997) Allowance for cost of claims in bonus-malus systems. ASTIN Bull 27:33–57CrossRef
43.
Pinquet J (1998) Designing optimal bonus-malus systems from different types of claims. ASTIN Bull 28:205–220CrossRefMATH
44.
Seshardi V (1993) The Inverse Gaussian Distribution. Oxford Science Publications, Oxford
45.
Shengwang M, Wei Y, Whitmore GA (1999) Accounting for individual overdispersion in a bonus-malus automobile insurance system. ASTIN Bull J IAA 29(2):327–337CrossRef
46.
Stone M (1977) An asymptotic equivalence of choice of model by cross-validation and Akaike’s criterion. J R Stat Soc Ser B 39:44–7MathSciNetMATH
47.
Trembley L (1992) Using the Poisson inverse Gaussian in bonus-malus systems. ASTIN Bull 22(1):97–106CrossRef
48.
49.
Tzougas G, Vrontos S, Frangos N (2015) Risk classification for claim counts and losses using regression models for location, scale and shape. Variance 9(1):140–157
50.
Tzougas G, Vrontos S, Frangos N (2018) Bonus-malus systems with two-component mixture models arising from different parametric families. N Am Actuar J 22(1):55–91MathSciNetCrossRefMATH
51.
52.
Yip K, Yau K (2005) On modeling claim frequency data in general insurance with extra zeros. Insur Math Econ 36:153–63CrossRefMATH
53.
Zamani H, Ismail N (2010) Negative binomial-Lindley distribution and its application. J Math Stat 6(1):4–9CrossRefMATH
54.
Zhou M, Li L, Dunson D, Carin L (2012) Lognormal and gamma mixed negative binomial regression. In: Proceedings of the... International Conference on Machine Learning. International Conference on Machine Learning (vol 2012, p 1343). NIH Public Access
Metadata
Title
The negative binomial-inverse Gaussian regression model with an application to insurance ratemaking
Authors
G. Tzougas
W. L. Hoon
J. M. Lim
Publication date
17-11-2018
Publisher
Springer Berlin Heidelberg
Published in
European Actuarial Journal / Issue 1/2019
Print ISSN: 2190-9733
Electronic ISSN: 2190-9741
DOI
https://doi.org/10.1007/s13385-018-0186-2

Other articles of this Issue 1/2019

European Actuarial Journal 1/2019 Go to the issue

Original Research Paper

Bivariate regular variation among randomly weighted sums in general insurance

Discussion on recent papers

Discussion on “The negative binomial-inverse Gaussian regression model with an application to insurance ratemaking” (by Tzougas et al.)

Original Research Paper

Generational transfers within the occupational pension system in Switzerland

Discussion on recent papers

Discussion on “Experience rating in the classic Markov chain life insurance setting: an empirical Bayes and multivariate frailty approach” (Furrer)

Original Research Paper

Practical aspects of the aggregation of two risks in the Solvency II standard formula

Original Research Paper

Optimal dividend payments for a two-dimensional insurance risk process