Automobile Insurance Pric-2

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Automobile insurance pricing with GeneralizedLinear Models

Mihaela DavidFaculty of Economics and Business Administration

“Alexandru Ioan Cuza” University of IasiIasi, Romania

Abstract - The fundamental purpose of insurance is to providefinancial protection, offering an equitable method of transferringthe risk of a contingent or uncertain loss in exchange forpayment. Considering that not all the risks are equal, aninsurance company should not apply the same premium for allinsured risks in the portfolio. A commonly method to calculatethe insurance premium is to multiply the conditional expectationof the claim frequency with the expected cost of claims. In thispaper, Generalized Linear Models are employed to estimate thetwo components of the premium given the observedcharacteristics of the policyholders. A numerical illustrationbased on the automobile insurance portfolio of a Frenchinsurance company is included to support this approach.

Keywords- insur ance pri cing, insurance premium, frequency ofclaims, cost of cl aims, General ized L in ear Models

I. I NTRODUCTION The fundamental role of insurance is to provide financial

safety and security against a possible loss on a particular event.The entire process of insurance consists in offering an equitablemethod of transferring the risk of a contingent or uncertain lossin exchange for payment. Considering that not all the risks are

equal, an insurance company should not apply the same premium for all insured risks in the portfolio. The necessity ofdifferent charging tariffs is emphasized by the insurance

portfolio heterogeneity that leads directly to the so-calledconcept of adverse selection. This basically presumes chargingsame tariff for the entire portfolio, meaning that theunfavourable risks are also assured (at a lower price) and as anadverse effect, it discourages insuring medium risks. The idea

behind non-life insurances pricing comes precisely in anattempt to combat the adverse selection. Therefore, it isextremely important for the insurer to divide the insurance

portfolio in sub-portfolios based on certain influence factors. Inthis way, the policyholders with similar risk profile will pay thesame reasonable insurance premium.

A usual method to calculate the premium is to find theconditional expectation of the claim frequency given the riskcharacteristics and to combine it with the expected cost ofclaims. The process of measurement and construction of a fairtariff structure is performed by the actuaries, who over time

proposed and applied different statistical models. In the contextof actuarial science, linear regression was employed to evaluatethe insurance premium. Considering the complexity of the

phenomenon to be modelled and some methodological aspectsrelated to the insurance data, the assessment of insurance

premium does not fit anymore in the framework of linear

regression. Antonio and Valdez [2] point out that, after decadesdominated by statistically unsophisticated models, it is nowrecognized that Generalized Linear Models (GLMs) constitutethe efficient tool for risk classification. Kaas, Goovaerts,Dhaene and Denuit [14] state that models allow the randomdeviations from the mean to have a distribution different fromthe normal and the mean of the random variable may be alinear function of the explanatory variables on some otherscale. GLMs allow modelling a non-linear behaviour and a

non-Gaussian distribution of residuals. This aspect is veryuseful for the analysis of non-life insurances, where claimfrequency and costs follows an asymmetric density that isclearly non-Gaussian. GLMs development has contributed toquality improvement of the risk prediction models and to the

process of establishing a fair tariff or premium given the natureof the risk.

This paper present an example based on real-life insurancedata in order to illustrate several techniques in the frameworkof GLMs. These illustrations are relevant for the insurers toimplement the used techniques in practice in order to obtainequitable and reasonable premiums. In this purpose, thestructure of the paper is as follows. Section 2 presents the basic

distributions that can be used to model the two components of pure premium, namely the frequency and cost of claims. In this part of paper, the reasons for using these distributions areexplained and a special test concerning the difference betweenclaim frequency models is also described. Section 3 isdedicated to an empirical application using a Frenchautomobile insurance portfolio. This is followed by adiscussion and an interpretation of the obtained results.Concluding remarks are summarized in Section 4.

II. METHODOLOGY APPROACH The methodological section of this paper aims to present

some specific issues related to the GLMs and the role of thesemodels within non-life insurance business. The main focus ison the definition, interpretation and presentation of the

properties and limits of the insurance premium calculationmodels.

A. Generalized Linear Models (GLMs)The implementation merits of Generalized Linear Models,

both in actuarial science and statistics, goes to British actuariesfrom City University, John Nelder and Robert Wedderburn. Inthe paper published in 1972, they demonstrate that thegeneralization of the linear modeling allows the deviation fromthe assumption of normality, extending the Gaussian model to

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a particular family of distribution, namely the exponentialfamily. Members belonging to this family include, but notlimited to, the Normal, Poisson, Binomial and Gammadistributions.

Nelder and Wedderburn [17] suggest that the estimation ofthe GLMs parameters to be performed through maximumlikelihood method, so that the parameter estimates are obtainedthrough an iterative algorithm. The contribution of Nelder indeveloping and completing the GLMs theory continues whilecollaborating with the Irish statistician Peter McCullach, whose

paper [16] offers detailed information on the iterative algorithmand the asymptotic properties of the parameter estimates.

Since the implementation of GLMs techniques, thecomplexity of papers is remarkable, many researchers succeedto highlight, develop or improve the assumptions imposed bythe practical application of these models in non-life insurance.Among the precursors of GLMs approach as the mainstatistical tool in determining the insurance tariffs is noted in[15]. Resorting to these models, he aims to estimate the

probability of risk occurrence in automobile insurance, toestablish the insurance premium and also to measure theeffectiveness of the models used to estimate it. Charpentier andDenuit [6] have a significant contribution in actuarial area,succeeding to cover in a modern perspective all the aspects ofinsurance mathematics. Boucher, Denuit and Guillen [4]

provide a comprehensive reference on several aspects of a priori risk modeling, with an emphasis on claim frequency.Frees [10] employs the main statistical regression models forinsurance, illustrating several case studies. Other usefulreference have pointed out the contribution of Jong and Zeller[13], Kaas, Goovaerts, Dhaene and Denuit [14] or Ohlsson andJohansen [18], who highlight the GLMs particularities in non-life insurance risk modeling.

GLMs are defined as an extension of the Gaussian linearmodels framework that is derived from the exponential family.The purpose of these models is to estimate an interest variable( ) depending on a certain number of explanatory variables( ). During the actuarial analysis, considering that theexogenous variables represent information about the insured orhis assets, the dependent variable can be one of the following:

a binary variable that can only have the value zero orone, the phenomenon studied in this case being the

probability of a risk occurrence, for which it appliesthe binomial regression models ( logit , probit and log-log complementary models );

a discrete variable, with values belonging to the set ofnatural numbers, while following the modelingfrequency of the risk occurrence. In this case thePoisson regression model will be applied;

a continuous variable, with values belonging to the setof positive real numbers, while following theeconometric analysis of the risk occurrence cost. In thiscase the Gamma regression model is considered.

Conditioned by the explanatory variables ( ), the randomvariables are considered to be independently, butnot identically distributed, that have the probability density

given by the following function, specializing to a probabilitydensity function in the continuous case and a probability massfunction in the discrete case as in [17]:

where represents a subassembly that belongs to or set, is the natural parameter and is the scale parameter. In

binomial and Poisson distributions, the scale parameter has thevalue 1, and for the Gamma distribution is unknown and hasto be estimated.

Similar with the Gaussian model approach, the purpose ofthe econometrical modeling is to obtain the expected values ofthe dependent variables through conditional means, givenindependent observations. In this case, the searched parameters

, allow writing a function ( ) for the mean ofthe variable as a linear combination of the exogenousvariables :

the monotonous and differentiable function is known as a

link function because it connects the linear predictor withthe mean .

Jong and Zeller [13] cover in a practical and rigorousmanner the standard exponential family distributions, focusingon issues related to insurance data and discussing all techniquesthat are illustrated on data sets relevant to insurance. As theobjective of this paper is to establish the insurance premium,further are introduced and detailed only the models employedto estimate the frequency and cost of claims.

B. Estimation models of claim frequency Poisson model

The statistical analysis of counts data, known in theeconometric literature as rare events, has a long and richhistory. The Poisson distribution was derived as a limiting caseof the binomial distribution by Poisson (1837) and exemplifiedlater by Bortkiewicz (1898) in the famous study regarding theannual number of deaths caused by the mules’ kicks in thePrussian army. Cameron and Trivedi [5] have an importantcontribution to the development of counts regression models.They have managed to highlight the particularities of Poissonregression approach in estimating the claim frequency as a

particular case of GLMs.

Within non-life insurance business, it has beendemonstrated that the usage of the GLMs techniques in order toestimate the frequency of claims, has an a priori Poissonstructure. In actuarial literature, the Poisson model is presentedas the modeling archetype of the “event counts” as in [2], alsoknown in insurance as the frequency of claims. In many papersas in [7, 8, 9, 11, 20], the Poisson model is considered the maintool for the estimation of claim frequency in non-life insurance.



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The discrete random variable (claim frequency orobserved number of claims), conditioned by the vector ofexplanatory variables (the insured characteristics), isassumed to be Poisson distributed. Therefore, for the insured ,the probability that the random variable takes the value , is given by the density:

The Poisson distribution implies a particular form of

heteroskedasticity, leading to the equidispersion hypothesis orthe equality of the mean and variance of claim frequency.Thus, the Poisson distribution parameter has a double meaning,representing at the same time the mean and the variancedistribution [3]:

The standard estimator for this model is the maximum

likelihood estimator. The likelihood function is defined asfollows:

∏ ∏

Using a logarithm in both sides of the previous equation, it

is obtained the log-likelihood function:

It can be easy verified that the first two partial derivatives

of the log-likelihood function exists and are expressed asfollows:

The maximum likelihood estimators are the solutions of

the previous likelihood equations that are obtained bydifferentiating the log-likelihood in terms of the regressioncoefficients and solving them to zero. The equations formingthe system are not generating explicit solutions and thereforethey have to be solved numerically using an iterative algorithm.As underlined in [6], the most common iterative methods areconsidered Newton-Raphson and Fisher information. Hilbe[12] explains at length that this type of algorithm functions byupdating the estimates based on the value of log-likelihoodfunction.

The main limit of Poisson model is that the equidispersionassumption is not generally respected in practice, leading tooverdispersion, meaning that the conditioned variance isgreater than the mean of claim frequency. One of the mostimportant implications of overdispersion is related to theunderestimation of the regression parameter which means thatsome risk factors could appear to be significant when actuallythey have no considerable influence on the variation of claim

frequency. In this regard, the literature presents the quasi-Poisson model as a several enhanced models in order to correctthe Poisson overdispersed data.

Quasi-Poisson model

McCullagh and Nelder [16], based on the data provided byLloyd's Register of Ships, apply the quasi-Poisson model toexplain the frequency of damages caused to the cargo ships.Allain and Brenac [1] sustain also the use of quasi-Poissonmodel in the presence of a high level of dispersion, arguing thatthe approach of Poisson model could imply the acceptance ofsome explanatory variables as apparently significant factorswhich in reality they do not have any important impact on thestudied phenomen.

In a road accidents study the overdispersion is modelledthrough the quasi- Poisson regression model, which involves a τdispersion parameter, describing the incompatibility betweenthe variance and the mean as in [1]:

The principle of this model is to estimate the regression

parameters to minimize the quasi-likelihood:

where is the deviance function of Poisson model

determined as follows:

(| ) As mentioned in McCullagh and Nelder [16], the

overdispersion parameter is estimated by equating thePearson statistic to the residual degrees of freedom asfollows:

̃ { }

where n represents the number of observations and p is thenumber of parameters from the regression model.

The parameter estimatates ( ) are identical to those for thePoisson model, which shows that estimates are identical, butthe standard errors of the estimators for the quasi-Poisson

model are modified by the dispersion factor . McCullaghand Nelder [16] demonstrate that the parameter estimatorsequality of the two models derived from the shape of thelikelihood function corresponding to Poisson distribution and



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The previous expression is being interpreted as anorthogonality relationship between explanatory variables andresiduals.

Actuarial literature argues that the main advantage ofapplying the Gamma model is due to parameters and ,through which more flexibility is obtained while estimating thecost of claims.

Gamma model’ s goodness of fitGamma regression model’s goodness of fit is performed by

means of Fisher statistics that is constructed on the basis of thedifference between the deviance of model without explanatoryvariables ( ), and the deviance of model that includes all thesignificant risk factors ( ). Considering the estimate of thedispersion parameter ( ) for the latter model, there is obtainedthrough the expression bellow:

that follows approximately the Fischer distribution of

parameters: is the number of parameters corresponding tomodel without variables; represents the number of

parameters from the model that includes the significant riskfactors and n is the number of observations used.

D. Pure premium calculationIn non-life insurance, the pure premium represents the

expected cost of all claims declared by policyholders during theinsured period. The calculation of the premium is based onstatistical methods that incorporate all available informationabout the accepted risk, thereby aiming at a more accurateassessment of tariffs attributed to each insured.

The basis for calculating the pure premium is theeconometric modeling of the frequency and cost of claimsdepending to the characteristics that define the insurancecontracts. The pure premium is the mathematical expectationof the annual cost of claims declared by the policyholders andis obtained by multiplying the two components, the estimatedfrequency and estimated cost of claims:

∑ for the claims amount independent of their

number .

Within the context of insurance pricing, the separateevaluation of frequency and cost of claims is particularlyrelevant since the risk factors that influence the twocomponents of the insurance premium are usually different.Essentially, the separate analysis of the two elements providesa clearer perspective on how the risk factors are influencing theinsurance tariff.

III. PREPARE YOUR PAPER BEFORE STYLING The empirical part of this paper includes a brief

presentation of the used data, based on which a numericalillustration of the described techniques is performed.

A. Data UsedIn this paper, the data used constitute a French automobile

portfolio insured against theft of the vehicle and possiblydamage to the vehicle, comprising 50000 polices registeredduring the year 2009. An insurance policy corresponds to one

policyholder and the elements included in the policies are theanalysis factors presented bellow. Hence, except the explainedvariables, the frequency and cost of claims , the other ones areconsidered risk factors, known a priori by the insurer and areused to customize the profile of each insured. These exogenousvariables reflect the insured characteristics: age (18-75 years),

profession (employed, housewife, retired, self-employed,unemployed); the vehicle features category (large, medium,small), brand (A, B, C, D, E, F) , GPS (Yes, No), purpose ofvehicle usage (private, professional); the insurance contractscharacteristics: duration (0-15 years), bonus-malus coefficient (50-150 by 10).

Among the explanatory variables introduced in the analysis,bonus-malus coefficient presents a particular interest, assumingthe increase or decrease of insurance premium depending onthe number of claims registered by an insured during areference period. Therefore, if the policyholder does not causeany responsible accident, he receives a bonus , meaning that theinsurance premium will be reduced with 5%. Contrary, if theinsured is responsible for the accident, he is penalized byapplying a malus of 25% for a claim declared, which will havethe consequence of a premium increase. The implementation

bonus-malus system is different from one country to another, but the principle remains the same, namely to purchase theencouragement of prudent insured and the discouragement ofthose who, for various reasons, declare many claims, andthereby they present a high degree of risk for the insurancecompany.

B. EquationsFurther there are presented and interpreted the results

obtained through the application of the models mentioned, based on which the pure premium is determined. The variablesentered previously are taken into consideration as risk factorsand the models are fitted using the SAS 9.3 software by meansof GENMOD procedure. This procedure enables the use ofType 3 analysis that allows the contribution assessment of eachrisk factor, considering all the others explanatory variables.The type 3 analysis provides the values of Chi-Square statisticsfor each variable by calculating two times the difference

between the log-likelihood of the model which includes all theindependent variables and the log-likelihood of the modelobtained by deleting one of the specified variables. This

statistics test appreciates the impact of each risk factor on thestudied phenomenon and follows the asymptotic distribution with df degrees of freedom, representing thenumber of parameters associated to the analyzed variable.

Poisson modelBy employing the Poisson model to estimate the frequency

of claims, the results obtained are shown in Table 1.



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TABLE I. LR STATISTICS FOR TYPE 3 A NALYSIS

SourcePoisson Regression(*) Poisson Regression(**)

Chi-Square Pr > ChiSq Chi-Square Pr > ChiSqAgeOccupCategBrandGPSBonusPoldurUse

87.9963.86

4.1646.9284.52

451.8035.55

1.25


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regarding the separate analysis of these two elements. Based onthe cost of claims, it is not possible to obtain conclusiveinformation regarding the risk occurrence probability and theinsurance company cannot properly divide the policyholders.

Nevertheless, the amount of cost is a fundamental componentconsidered while establishing the insurance premium.

The last step of claim cost analysis consist in measuring thequality of Gamma regression model by means of Fisherstatistic test detailed previously in the paper. The resultsobtained are shown in Table 3. Within the studied portfolio, forthe final regression model, the obtained value of Fisher statistictest ( = 98.445) is much higher than the theoretical value( = 1.831), meaning that the proposed Gammamodel fits well the data and its employment is significant inorder to explain the variation of claim cost.

Pure premium model The process of establishing the insurance premium resides

in using the same procedure GENMOD as noticed in previouscases, the obtained results being summarized in Table V. Inthis stage of non-life insurance pricing, the explained variable

is the product between the estimated frequency and theestimated cost of claims:

the calculated value representing the insurance pure premium established for insured , characterized by thevariables vector .

TABLE V. A NALYSIS OF P ARAMETER E STIMATES

Parameter

PoissonRegression

GammaRegression

PurePremiumEstimate

StdError Estimate

StdError

Intercept -0.301 0.074 8.456 0.108 6.268Age ( ) -0.043 0.001 -0.012 0.002 -0.031Occup - employed ( ) -0.336 0.036 -0.167 0.064 -0.433Occup - housewife ( ) -0.411 0.043 0.024 0.077 -0.394Occup - retired ( ) -0.045 0.065 0.023 0.111 -0.220Occup - self-employed ( )

-0.015 0.038 0.297 0.069 0.349

Brand - A ( ) -0.356 0.055 -0.421 0.093 -0.778Brand - B ( ) -0.357 0.057 -0.446 0.096 -0.875Brand - C ( ) -0.308 0.060 -0.300 0.103 -0.674Brand - D ( ) -0.112 0.056 -0.172 0.096 -0.359Brand - E ( ) -0.039 0.060 -0.203 0.102 -0.246GPS - no ( ) 0.179 0.029 - - 0.422Bonus-Malus ( ) 0.007 0.001 - - 0.008Poldur ( ) -0.025 0.003 - - -0.029

Considering this relationship within the analyzed insurance portfolio, the pure premium for each category of policyholdersis established based on the Gamma regression model, includingall the statistical relevant tariff variables that explain thevariation of claim frequency and costs. More explicit, therelation between the premium and the risk factors is expressthrough the regression model written as follows:

Therefore, this regression model allows to obtain the pure premium corresponding to each tariff class through theexpression: . For example, by using these results, itcan be established the higher risk profile of policyholder.Taking into consideration the coefficients’ sign, it can beobserved that the higher risk profile is presented by the

policyholders aged 18 years, being self-employed, with avehicle of brand F, without a GPS device, with a bonus-malusof 150 and being the client of the company for less than a year.

In summary, the obtained results lead to using tariffscorresponding to the proper risk levels induced by the insuredto the insurance company. The default purpose of the non-lifeinsurance pricing is deduced from the idea that the new policeswill be mostly established for drivers that fit the profilegenerated after establishing the insurance tariffs. Thereupon,the pure premium will be used for the new policyholders thatwill be classified in one of the tariff categories already defined

by the insurance company.

IV. CONCLUSIONS In this paper, it was considered an analysis of Generalized

Linear Models in order to establish the pure premium given thecharacteristics of the policyholders. Therefore, as a first stage,while using the Poisson and quasi-Poisson models in theframework of GLMs, it was obtained a decrease of claim

frequency along with an increase of insured’s age and age ofthe insurance contracts , and also an increase along with thebonus-malus coefficient increase. These results are consistentwith the reality of the studied phenomenon, so theirinterpretation is considered to be logic and valid.

After the comparison of these two regression models, weobserved that although the quasi-Poisson model corrects theoverdispersion, the risk factors included in Poisson regressionappear to be significant for both models. Therefore, theregression coefficients do not change and there is no changethat should be made in terms of establishing the expectedfrequency of claims. In the next analysis stage, by using theGamma regression model we obtained the estimated averagelevel of the cost of claims corresponding to each category of

policyholders.

Eventually, the empirical results have shown that for thenew customers, the insurance premium will be establishedconsidering a series of risk factors, like age , profession , brand ,

purpose of vehicle usage , GPS , bonus-malus coefficient andage of the insurance contract . Based on the regressioncoefficients’ sign, there could be established the profile of theriskier policyholders. Taking into consideration these elements,the insurance company can establish a fair and reasonable

premium associated with each insured profile. Moreover, the



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company could implement a pricing policy that could fairlydiscriminate the portfolio, and thereby allowing a betterunderstanding of insured’s behavior and an accurateassessment of the risks to be insured.

The conclusions of this study are representative and usefulfor the insurance company business, but they do not present ageneralized character, therefore they cannot be applied to all

portfolios or insurance companies. On one side, this aspect is justified by the data used and the risk factors considered duringthe analysis process, meaning that every insurer can usedifferent information on the insured to their benefit. On theother side, the used data is not obtained through a randomselection related to the entire population of policyholders.

ACKNOWLEDGMENT

This work was supported by the European Social Fundthrough Sectoral Operational Programme Human ResourcesDevelopment 2007-2013, project number POSDRU/159/1.5/S/

34 97, project title “Performance and Excellence in Doctoraland Postdoctoral Research in Economic Sciences Domain inRomania”.

R EFERENCES [1] E. Allain and T. Brenac, “Modèles linéaires généralisés appliqués à

l'étude des nombres d'accidents sur des sites routiers: le modèle dePoisson et ses extensions ”, Recherche Transports Sécurité, vol. 72, pp.3-18, 2012.

[2] K. Antonio and E.A. Valdez, “Statistical concepts of a priori and a posteriori r isk classification in insurance”, Advances in StatisticalAnalysis, vol. 96(2), pp. 187-224, 2012.

[3] L. Asandului, Metode statistice de analiză a datelor categoriale, WoltersKluwer, Bucureşti , 2010.

[4] J.P. Boucher, M. Denuit and M. Guillen, “Risk classification for claimscounts - A comparative analysis of various zero-inflated mixed Poissonand hurdle models ”, North American Actuarial Journal, vol. 11(4), pp.110-131, 2007.

[5] A.C. Cameron and P.K. Trivedi, Regression Analysis of Count Data,Econometric Society Monograph. New York: Cambridge UniversityPress, 1998.

[6] A. Charpentier and M. Denuit, Mathématiques de l’assurance non -vie,Tome II: Tarification et provisionnement. Paris: Economica, 2005.

[7] M. Denuit and S. Lang, “Nonlife ratemaking with Bayesian GAM’s” ,Insurance: Mathematics and Economics, vol. 35(3), pp. 627-647, 2004.

[8] G. Dionne and C. Vanasse, “A generalization of automobilemobileinsurance rating models: the negative binomial distribution with a

regression component ”, ASTIN Bulletin, vol. 19(2), pp. 199-212, 1989.[9] G. Dionne and C. Vanasse, “Automobile insurance ratemaking in the

presence of asymmetrical information ”, Journal of AppliedEconometrics, vol. 7(2), pp. 149-65, 1992.

[10] E.W. Frees, Regression Modeling with Actuarial and FinancialApplications. New York: Cambridge University Press, 2010.

[11] C. Gourieroux and J. Jasiak, “Heterogeneous INAR(1) model withapplication to car insurance ”, Insurance: Mathematics and Economics,vol. 34(2), pp. 177-192, 2004.

[12] Hilbe, J.M., 2014. Modeling count data. New York: CambridgeUniversity Press.

[13] P. Jong and G. Zeller, Generalized Linear Models for Insurance Data. New York: Cambridge University Press, 2008.

[14] R. Kaas, M. Goovaerts, J. Dhaene and M. Denuit, Modern ActuarialRisk Theory. London: Springer Verlag, 2009.

[15] L. Lemaire, “Automobile Insurance: Actuarial Models ”, HuebnerInternational Series on Risk, Insurance and Economic Security, 1985.

[16] P. McCullagh and J.A. Nelder, Generalized Linear Models, 2nd ed.London: Chapman and Hall, 1989.

[17] J.A. Nelder and R.W.M. Wedderburn, “Generalized linear interactivemodels ”, Journal of the Royal Statistical Society, pp. 370-384, 1972.

[18] E. Ohlsson and B. Johansson, Non-life insurance pricing withGeneralized Linear Models. London: Springer Verlag, 2010.

[19] J. Pinquet, “Allowance for cost of claims in bonus-malus systems ”,ASTIN Bulletin, vol. 27, pp. 33-57,

[20] K. Yip and K. Yau, “On modeling claim frequency data in generalinsurance with extra zeros ”, Insurance: Mathematics and Economics,vol. 36(2), pp. 153-163, 2005.



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