A k-Inflated Negative Binomial Mixture Regression Model: Application to Rate--Making Systems

TL;DR

This paper proposes a flexible k-Inflated Negative Binomial mixture regression model for insurance claim frequency analysis, demonstrating its effectiveness in designing fairer rate-making systems compared to traditional models.

Contribution

It introduces a novel k-Inflated Negative Binomial mixture regression model and applies it to insurance data, showing improved premium fairness over existing models.

Findings

The new model provides more equitable premiums for policyholders.

It outperforms traditional models in modeling claim frequency.

The model is effective when claim counts are uniformly distributed in past data.

Abstract

This article introduces a k-Inflated Negative Binomial mixture distribution/regression model as a more flexible alternative to zero-inflated Poisson distribution/regression model. An EM algorithm has been employed to estimate the model's parameters. Then, such new model along with a Pareto mixture model have been employed to design an optimal rate--making system. Namely, this article employs number/size of reported claims of Iranian third party insurance dataset. Then, it employs the k-Inflated Negative Binomial mixture distribution/regression model as well as other well developed counting models along with a Pareto mixture model to model frequency/severity of reported claims in Iranian third party insurance dataset. Such numerical illustration shows that: ({\bf 1}) the k-Inflated Negative Binomial mixture models provide more fair rate/pure premiums for policyholders under a rate--making system; and ({\bf 2}) in the situation that number of reported claims uniformly distributed in past experience of a policyholder (for instance $k_1=1$ and $k_2=1$ instead of $k_1=0$ and $k_2=2$). The rate/pure premium under the k-Inflated Negative Binomial mixture models are more appealing and acceptable.

Equations

$$P(Y=y|\mathbf{\theta })$$

$$E(Y)$$

$$M_Y(t)$$

$$F_Y(r)$$

$$P\left(Y=y\right)=p{I}_{\{k\}}(y)+q\sum _{j=1}^m{\phi }_j\left(\genfrac{}{}{0ex}{}{y+{\alpha }_j-1}{y}\right){(\frac{{\tau }_j}{1+{\tau }_j})}^{{\alpha }_j}{(\frac{1}{1+{\tau }_j})}^y.$$

$$\log \left({\lambda }_{il}\right)={\beta }_{0il}+\sum _{k=1}^p{\beta }_{kil}{x}_{ki}+{ϵ}_i,$$

$$f_{U_i}(u_i)=\sum _{j=2}^m{\phi }_j\frac{{u_i}^{{\alpha }_j-1}{{\alpha }_j}^{{\alpha }_j}{e}^{-{\alpha }_j{u}_i}}{\Gamma ({\alpha }_j)},$$

$$P(Y_{il}=y_{il}|\theta )$$

$$P\left(Y_{il}=y_{il}\right)$$

$$l_c\left(\mathbf{\theta }|y_i,{\mathbf{v}}_i,X_i\right)$$

$${\hat{v}}_{ij}^{(r+1)}$$

$$\mathbf{Q}$$

$$\frac{\partial Q_1}{\partial {\omega }_j}$$

$$\frac{\partial Q_2}{\partial B_j}$$

$$\frac{\partial Q_2}{\partial {\alpha }_j}$$

$${\hat{\omega }}_j^{\left(r+1\right)}$$

$${\hat{B}}_j^{\left(r+1\right)}$$

$${\hat{\alpha }}_j^{\left(r+1\right)}$$

$$l_c^{\left(r+1\right)}$$

$$f_Z(z|\mathbf{\vartheta })$$

$$\log \left({\theta }_{il}\right)=d_{0il}+\sum _{k=1}^p{d}_{kil}{w}_{ki}+{ϵ}_i,$$

$$f_{U_i}(u_i)=\sum _{j=1}^m{\rho }_j\frac{({\alpha }_j-1{)}^{{\alpha }_j}{u}_i^{-{\alpha }_j-1}}{\Gamma ({\alpha }_j)}{e}^{-({\alpha }_j-1)/u_i},$$

$$f_{Z_{il}|\vartheta }(z_{il})$$

$${\widehat{Rate}}_i(t+1)=e^{C_i{B}_{it+1}}\frac{{\int }_0^{\infty }{u}_i{\prod }_{l=1}^t{h}_{il}(u_i){\sum }_{j=1}^m{k}_j(u_i)d_{u_i}}{{\int }_0^{\infty }{\prod }_{l=1}^t{h}_{il}(u_i){\sum }_{j=1}^m{k}_j(u_i)d_{u_i}},$$

$$f_{{\Lambda }_{it+1}|({\mathbf{Y}}_i,{\mathbf{C}}_i)}({\lambda }_{it+1})$$

$${\widehat{Rate}}_i(t+1)={\int }_0^{\infty }{e}^{C_i{B}_{it+1}}{u}_i{f}_{{\Lambda }_{it+1}|({\mathbf{Y}}_i,{\mathbf{C}}_i)}({\lambda }_{it+1})d{u}_i\square$$

$${\widehat{Rate}}_i(t+1)$$

$${\widehat{Rate}}_i(t+1)=\frac{{\int }_0^{\infty }{\lambda }_{it+1}{\prod }_{l=1}^t\left(p{I}_{\left(y_{il}=k\right)}+q\frac{e^{-{\lambda }_{it+1}}{\left({\lambda }_{it+1}\right)}^{y_{il}}}{y_{il}!}\right){\sum }_{j=1}^m{\phi }_j\frac{{{\lambda }_{it+1}}^{{\alpha }_j-1}{{\tau }_j}^{{\alpha }_j}{e}^{-{\tau }_j{\lambda }_{it+1}}}{\Gamma ({\alpha }_j)}{d}_{{\lambda }_{it+1}}}{{\int }_0^{\infty }{\prod }_{l=1}^t\left(p{I}_{\left(y_{il}=k\right)}+q\frac{e^{-{\lambda }_{it+1}}{\left({\lambda }_{it+1}\right)}^{y_{il}}}{y_{il}!}\right){\sum }_{j=1}^m{\phi }_j\frac{{{\lambda }_{it+1}}^{{\alpha }_j-1}{{\tau }_j}^{{\alpha }_j}{e}^{-{\tau }_j{\lambda }_{it+1}}}{\Gamma ({\alpha }_j)}{d}_{{\lambda }_{it+1}}}.$$

$${\widehat{BP}}_i^{t+1}=e^{W_i{D}_{it+1}}\frac{{\sum }_{j=1}^m{\varphi }_j\frac{({\eta }_j-1{)}^{{\eta }_j}}{\Gamma ({\eta }_j)}\frac{\Gamma ({\eta }_j+K_i-1)}{({\eta }_j+{\sum }_{l=1}^t{\sum }_{s=1}^{k_{il}}\exp (-W_i{D}_{il})z_{ils}-1{)}^{{\eta }_j+K_i-1}}}{{\sum }_{j=1}^m{\varphi }_j\frac{({\eta }_j-1{)}^{{\eta }_j}}{\Gamma ({\eta }_j)}\frac{\Gamma ({\eta }_j+K_i)}{({\eta }_j+{\sum }_{l=1}^t{\sum }_{s=1}^{k_{il}}\exp (-W_i{D}_{il})z_{ils}-1{)}^{{\eta }_j+K_i}}},$$

$$f_{{\Theta }_{it+1}|({\mathbf{Z}}_i,{\mathbf{W}}_i)}({\theta }_{it+1})$$

$${\widehat{BP}}_i^{t+1}={\int }_0^{\infty }{e}^{W_i{D}_{it+1}}{u}_i{f}_{{\Theta }_{it+1}|({\mathbf{Z}}_i,{\mathbf{W}}_i)}({\theta }_{it+1})d{u}_i.\square$$

$${\widehat{BP}}_i^{t+1}=\frac{{\sum }_{j=1}^m{\varphi }_j\frac{({\eta }_j-1{)}^{{\eta }_j}}{\Gamma ({\eta }_j)}\frac{\Gamma ({\eta }_j+K_i-1)}{({\eta }_j+{\sum }_{l=1}^t{\sum }_{s=1}^{k_{il}}{z}_{ils}{)}^{{\eta }_j+K_i-1}}}{{\sum }_{j=1}^m{\varphi }_j\frac{({\eta }_j-1{)}^{{\eta }_j}}{\Gamma ({\eta }_j)}\frac{\Gamma ({\eta }_j+K_i)}{({\eta }_j+{\sum }_{l=1}^t{\sum }_{s=1}^{k_{il}}{z}_{ils}{)}^{{\eta }_j+K_i}}}.$$

$$P(Y=y|\mathbf{\theta })$$

Full text

**A k-Inflated Negative Binomial Mixture Regression Model: Application to Rate–Making Systems

**Amir T. Payandeh Najafabadia,111Corresponding author: amirtpayandeh@sbu.ac.ir & Saeed MohammadPourb

a Department of Mathematical Sciences, Shahid Beheshti University, G.C. Evin, 1983963113, Tehran, Iran.

b E.C.O College of Insurance, Allameh Tabatabái University, Tehran, Iran.

**Abstract

**

This article introduces a k-Inflated Negative Binomial mixture distribution/regression model as a more flexible alternative to zero-inflated Poisson distribution/regression model. An EM algorithm has been employed to estimate the model’s parameters. Then, such new model along with a Pareto mixture model have been employed to design an optimal rate–making system. Namely, this article employs number/size of reported claims of Iranian third party insurance dataset. Then, it employs the k-Inflated Negative Binomial mixture distribution/regression model as well as other well developed counting models along with a Pareto mixture model to model frequency/severity of reported claims in Iranian third party insurance dataset. Such numerical illustration shows that: (1) the k-Inflated Negative Binomial mixture models provide more fair rate/pure premiums for policyholders under a rate–making system; and (2) in the situation that number of reported claims uniformly distributed in past experience of a policyholder (for instance $k_{1}=1$ and $k_{2}=1$ instead of $k_{1}=0$ and $k_{2}=2$). The rate/pure premium under the k-Inflated Negative Binomial mixture models are more appealing and acceptable.

Keywords: Negative Binomial regression; Poisson regression; Mixture model; Overdispersed behavior; Heavy–tail behavior; Inflated model; EM algorithm; Rate–making System.

2010 Mathematics Subject Classification: 62Jxx, 91B30, 97M30

1 Introduction

Modeling count data is an interesting topic in variety fields of applied sciences, such as actuarial sciences, economics, sociology, engineering, etc. In many practical situation the popular classical Poisson regression model fails to model count data which exhibit overdispersion (i.e., the variance of the response variable exceeds its mean). Moreover, strict assumptions on Poisson distribution make it more less applicable in situation that such assumption cannot be strictly verified. The Negative Binomial distribution/regression has become more and more popular as a more flexible alternative to Poisson distribution/regression. In a situation that strict requirements for Poisson distribution cannot be verified, the Negative Binomial distribution is an appropriate choice (Johnson et al., 2005). Moreover, the Negative Binomial is an appropriate choice for overdispersed count data that are not necessarily heavy–tailed (Aryuyuen & Bodhisuwan, 2013).

For count data, the overdispersed behavior has been arrived by either observing excess of a single value more than number of expected under the model or the target population consisting of several sub-populations. Using k-Inflated and mixture models are two popular statistical approach to dealing with an overdispersed behavior. Simar (1976) and Laird (1978) were two authors who employed Poisson mixture models to considering an overdispersed behavior. Lambert (1992) considered a zero-inflated Poisson regression model to take into account an overdispersed behavior. Wedel et al. (1993) Brännaäs & Rosenqvist (1994), Wang et al. (1996), Alfó & Trovato (2004), Wang et al. (1998), among others, developed idea of using a finite mixture Poisson regression model to handel overdispersion.

Greene (1994) and Hall (2000) were pioneer authors who employed zero-inflated Negative Binomial regression to model overdispersion. The ordinary Negative Binomial distribution can be viewed as a mixture of Poisson and gamma distributions (Simon, 1961). To handel an overdispersion phenomena, several extension of Negative Binomial distribution have been introduced by authors. For instance Negative Binomial exponential distribution (Panjer & Willmot, 1981), Negative Binomial Pareto distribution (Meng et al., 1999), Negative Binomial Inverse Gaussian distribution (Gómez-Déniz et al., 2008), Negative Binomial Lindley distribution (Zamani & Ismail, 2010), Negative Binomial Beta Exponential distribution (Pudprommarat, 2012), and Negative Binomial Generalized Exponential distribution (Pudprommarat et al., 2012).

In 2014, Lim et al. considered a k-Inflated Poisson mixture model which simultaneously takes into account both inflated and mixture approaches to handel an overdispersion phenomena. Moreover Tzougas et al. (2014)’ introduced a Negative Binomial mixture model to model an overdispersion phenomena. This article follows Lim et al. (2014)’s and generalized Tzougas et al. (2014)’s findings. More precisely, It introduces a k-Inflated Negative Binomial mixture distribution/regression. To show practical application of our finding, we consider the problem of designing an optimal rate–making system. Then, premium of such optimal rate–making system has been evaluated using the result of this article.

This article has been structured as follows. The k-Inflated Negative Binomial mixture model, some of its properties, and an EM algorithm, to estimated its parameters, have been developed in Section 2. The Pareto mixture regression model has been given in Section 3. Application of the k-Inflated Negative Binomial mixture model along with a Pareto mixture model to design an optimal rate–making system have been given in Section 4. Section 5 employs our model, as well as other well-known model, to evaluate rate, base, and pure premiums under a rate–making system for Iranian third party insurance dataset. Base upon three comparison methods, Section 6 shows that our model provides more accurate (in some sense) results. Section 7 employs our well fitted models to calculate rate and pure premiums under two different Scenarios. Conclusion and suggestions have been given in Section 8

2 k-Inflated Negative Binomial mixture regression model

The k-Inflated Negative Binomial mixture, say kINBM, distribution arrives by combining $m$ weighted mixture Negative Binomial distribution with a single mass at point $k.$ The probability mass function for a kINBM distribution has been given by

[TABLE]

where $k\in{\mathbb{N}}$ and ${\boldsymbol{\theta}}$ stands for all $3m$ unknown parameters. Moreover, $\sum_{j=1}^{m}p_{j}=1$ and $p_{j},\alpha_{j},\tau_{j}\geq 0,$ for all $j=1,\cdots,m.$ By a straightforward calculation, one may show that

[TABLE]

where $RIBeta_{x}(a;b)=\int_{0}^{x}t^{a-1}(1-t)^{b-1}dt\Gamma(a+b)/(\Gamma(a)\Gamma(b)),$ for $a,b\geq 0,x\in[0,1],$ stands for the regularized incomplete beta function.

It is well-known that a Negative Binomial distribution can be arrived by mixing two Poisson and gamma distributions (Simon, 1961). The following generalized the above fact to the kINBM distribution.

Corollary 1**.**

Suppose random variable $Y,$ given parameter $\Lambda=\lambda,$ has been distributed according to a k-Inflated Poisson distribution with probability mass function $P(Y=y|\lambda)=pI_{\{k\}}(y)+q\exp(-\lambda_{i})(\lambda_{i})^{y}/y!,$ where $p~{}\&~{}q\in[0,1]$ and $p+q=1.$ Moreover, suppose that parameter $\lambda$ has been distributed according to a finite mixture gamma distribution $f_{\Lambda}(\lambda)=\sum^{m}_{j=1}{{\varphi}_{j}{\lambda}^{{\alpha}_{j}-1}{{\tau}_{j}}^{{\alpha}_{j}}e^{-{\tau}_{j}\lambda}/\mathit{\Gamma}({\alpha}_{j})},$ where, for all $j=1,\cdots,m,$ ${\varphi}_{j}\in[0,1]$ and $\sum^{m}_{j=1}{{\varphi}_{j}}=1.$ Then, unconditional distribution of $Y$ has a kINB finite mixture distribution with probability mass function

[TABLE]

For practical application, in Equation (2), we set $q:=\sum_{s=1}^{m}\omega_{s}$ and ${\varphi}_{j}:=\omega_{j}/\sum_{s=1}^{m}\omega_{s}.$

Now, to formulated a kINBM regression model, suppose that for an $i^{\hbox{th}}$ individual, information on count response variables $Y_{i1},\cdots,Y_{it}$ along with information on $p$ covariates $X_{1},\cdots,X_{p}$ are available. Also suppose that $Y_{il}$ given parameter $\Lambda_{il}=\lambda_{il}$ has been distributed according to a k-Inflated Poisson distribution with probability mass function $P(Y_{il}=y_{il}|\lambda_{il})=pI_{\{k\}}(y_{il})+q\exp(-\lambda_{il})(\lambda_{il})^{y}_{il}/y_{il}!,$ where $p~{}\&~{}q\in[0,1]$ and $p+q=1.$ Moreover, suppose that parameter $\lambda_{il}$ can be evaluated by the following regression model

[TABLE]

where $\beta_{0i},\cdots,\beta_{pi}$ are regression coefficients and $u_{i}=\exp(\epsilon_{i})$ has been distributed according to a finite mixture gamma distribution with density function

[TABLE]

where $\sum^{m}_{j=2}{\varphi}_{j}=1$ and ${\alpha}_{j}\geq 0$. To have $E(\epsilon_{i})=0,$ we set both parameters of all gamma distributions, in the finite mixture gamma distribution, to be equal.

Using the law of total probability and setting $d_{il}:=\beta_{0il}+\sum_{k=1}^{p}\beta_{kil}x_{ki},$ one may show that

[TABLE]

where $\theta$ stands for all unknown parameters. Now by setting $p=1/(1+\sum^{m}_{s=2}{e^{\omega_{s}}}),$ $q{\varphi}_{j}=e^{\omega_{j}}/(1+\sum^{m}_{s=2}{e^{\omega_{s}}}),$ for $j=2,\dots,m,$ and $\left(\genfrac{}{}{0.0pt}{}{y_{il}+{\alpha}_{j}-1}{y_{il}}\right):=\frac{\Gamma(y_{il}+\alpha_{j})}{y_{il}!\Gamma(\alpha_{j})},$ the kINBM regression model can be restated as

[TABLE]

where $t_{ilj}:=\alpha_{j}/(\alpha_{j}+\exp\{X_{i}B_{il}\})$ and $X_{i}B_{il}:=\beta_{0il}+\sum_{k=1}^{p}\beta_{kil}x_{ik}.$

Parameters estimation

All unknown parameters of the kINBM regression (2) can be represented as ${\boldsymbol{\theta}}:=({\boldsymbol{\omega}},{\boldsymbol{\alpha}},{\boldsymbol{B}}).$ Now to provide a Maximum likelihood estimator, say MLE, for ${\boldsymbol{\theta}},$ one may employ an EM algorithm. In statistical literature, the EM algorithm is a well-known and practical method to obtain the Maximum likelihood estimators for parameters in an arbitrary finite mixture model (McLachlan & Krishnan, 1997). Now suppose that number of components, $m,$ is given, and ${\boldsymbol{\ }\boldsymbol{v}}_{i}=(v_{i1},\cdots,v_{im})$ stands for the latent vector of component indicator variables, where for $i=1,\cdots,n$ and $j=1,\cdots,m,$ $v_{ij}=1$ whenever observation $i$ comes from $j^{\hbox{th}}$ component and $v_{ij}=0,$ otherwise. Therefore, we assume that each observation has been arrived from one of the $m$ components, but the component it belongs to is unobservable and therefore considered to be the missing data.

Now using the Multinomial distribution for the unobservable vector ${\boldsymbol{\ }\boldsymbol{v}}_{i},$ the complete data loglikelihood function, for, the kINB regression model, can be written as the following, see Rigby & Stasinopoulos (2009) for an update information.

[TABLE]

where ${\boldsymbol{\theta}}:=({\boldsymbol{\omega}},{\boldsymbol{\alpha}},{\boldsymbol{B}})$ stands for all unknown parameters, $t_{j}:=\alpha_{j}/(\alpha_{j}+\exp\{X_{i}B_{j}\}),$ and $X_{i}B_{j}:=\beta_{0j}+\sum_{k=1}^{p}\beta_{kj}x_{ik}.$

The EM algorithm employs the following two steps to maximize the above loglikelihood function.

E-step:

In this step, using given data along with current estimates ${\widehat{\boldsymbol{\theta}}}^{(r)}:=({\widehat{\boldsymbol{\omega}}}^{(r)},{\widehat{\boldsymbol{\alpha}}}^{(r)},{\widehat{\boldsymbol{B}}}^{(r)})$ obtained from the $r^{\hbox{th}}$ iteration, the probability ${\hat{v}}_{ij}$ estimates. This probability, $(r+1)^{\hbox{th}}$ iteration, can be stated as

[TABLE]

where ${\widehat{\boldsymbol{t}}}^{(r)}_{j}:={\widehat{\alpha}}^{(r)}_{j}/({\widehat{\alpha}}^{(r)}_{j}+\exp\{X_{i}{\widehat{B}}^{(r)}_{j}\}),$ and $X_{i}{\widehat{B}}^{(r)}_{j}:={\widehat{\beta}}^{(r)}_{0j}+\sum_{k=1}^{p}{\widehat{\beta}}^{(r)}_{kj}x_{ik}.$

M-step:

Given the probability $\hat{v}_{ij},$ this step maximizes, in the $(r+1)^{\hbox{th}}$ iteration, the following loglikelihood $Q(\cdot)$ with respect to ${\boldsymbol{\theta}}:=({\boldsymbol{\omega}},{\boldsymbol{\alpha}},{\boldsymbol{B}}).$

[TABLE]

Updated parameters ${\widehat{\boldsymbol{\theta}}}^{(r+1)}:=({\widehat{\boldsymbol{\omega}}}^{(r+1)},{\widehat{\boldsymbol{\alpha}}}^{(r+1)},{\widehat{\boldsymbol{B}}}^{(r+1)})$ have been arrived by solving the following equation.

[TABLE]

Since the above three equations cannot solve explicitly, such updated parameters have been obtained using the following Iteratively Reweighted Least Squares, say IRLS, method.

[TABLE]

In IRLS method, $E\left(-{\partial}^{2}\left({\mathrm{log}\mathrm{likelihood}\ }\right)/\partial{\mathrm{parameter}}^{\mathrm{2}}\right)$ can be viewed as the Fisher information matrix and $\partial\left({\mathrm{log}\mathrm{likelihood}\ }\right)/\partial\mathrm{parameter}$ as score function.

After updated Parameter estimates ${\widehat{\boldsymbol{\theta}}}^{(r+1)}:=({\widehat{\boldsymbol{\omega}}}^{(r+1)},{\widehat{\boldsymbol{\alpha}}}^{(r+1)},{\widehat{\boldsymbol{B}}}^{(r+1)}),$ the complete data loglikelihood for $(r+1)^{\hbox{th}}$ iteration, arrives by

[TABLE]

where ${\widehat{\boldsymbol{t}}}^{(r+1)}_{j}:={\widehat{\alpha}}^{(r+1)}_{j}/({\widehat{\alpha}}^{(r+1)}_{j}+\exp\{X_{i}{\widehat{B}}^{(r+1)}_{j}\}),$ and $X_{i}{\widehat{B}}^{(r+1)}_{j}:={\widehat{\beta}}^{(r+1)}_{0j}+\sum_{k=1}^{p}{\widehat{\beta}}^{(r+1)}_{kj}x_{ik}.$ Now, in the E-step $v_{ij}-$s have been estimated. This loop has been repeated until the difference $\left|l^{\left(r+1\right)}_{c}-l^{\left(r\right)}_{c}\right|$ has been converged, in some sense.

It is worthwhile to mention that, since regression coefficients ${\boldsymbol{B}}=(\beta_{0},\cdots,\beta_{p})^{\prime}$ have been estimated using the MLE methods. therefore, number of mixture component impact on such estimators.

3 Pareto mixture regression model

The Pareto mixture distribution arrives by combining $m$ weighted mixture Pareto distributions. The density function for a Pareto mixture distribution has been given by

[TABLE]

where ${\boldsymbol{\vartheta}}=({\boldsymbol{\rho}},{\boldsymbol{\alpha}},{\boldsymbol{\gamma}})$ stands for all $3m$ unknown parameters. Moreover, $\sum_{j=1}^{m}\rho_{j}=1$ and $\rho_{j},\alpha_{j}\geq 0,$ for all $j=1,\cdots,m.$ More details on this distribution can be found in Tzougas et al. (2014).

Tzougas et al. (2014) showed that a Pareto mixture distribution can be arrived by mixing two exponential and inverse gamma distributions.

Now, to formulated a Pareto mixture regression model, suppose that for an $i^{\hbox{th}}$ individual, information on continuous response variables $Z_{i1},\cdots,Z_{it}$ along with information on $p$ covariates $W_{1},\cdots,W_{p}$ are available. Also suppose that $Z_{il}$ given parameter $\Theta_{il}=\theta_{il}$ has been distributed according to an exponential distribution with density function $f_{Z_{il}|\Theta_{il}=\theta_{il}}(z_{il})=\exp\{-z_{il}/\theta_{il}\}/\theta_{il}.$ Moreover, suppose that parameter $\theta_{il}$ can be evaluated by the following regression model

[TABLE]

where $d_{0i},\cdots,d_{pi}$ are regression coefficients and $u_{i}=\exp(\epsilon_{i})$ has been distributed according to a finite mixture Inverse gamma distribution with density function

[TABLE]

where $\sum^{m}_{j=1}\rho_{j}=1$ and $\rho_{j},~{}{\alpha}_{j}\geq 0$. To have $E(\epsilon_{i})=0$ in Equation (6) we set $\gamma_{j}=\alpha_{j}-1,$ for $j=1,\cdots,m.$

Using the law of total probability and setting $b_{il}:=d_{0il}+\sum_{k=1}^{p}d_{kil}w_{ki},$ one may show that

[TABLE]

Similar to the kINB regression/distribution the maximum liklihood estimator for parameters of a Pareto mixture regression/distribution can be obtained using the EM algorithm. Fortunately, Rigby & Stasinopoulos (2001) developed a R package, named ’GAMLSS‘, for such propose, see Rigby & Stasinopoulos (2001, 2009) for more details.

4 Application to posteriori rate–making system

The rate–making system is a non-life actuarial system which rates policyholders based upon their last $t$ years record (Payandeh Najafabadi et al., 2015). A rate–making system based upon policyholders’ characteristics assigns a priori premium for each policyholder. Then, it employs the last $t$ years claims experience of each insured to update such priori premium and provides posteriori premium (Boucher & Inoussa, 2014). The Bonus–Malus system is a commercial and practical version of the rate–making system which takes into account current year policyholders’ experience to determine their next year premium.

There is a considerable attention from authors to study rate–making systems (or Bonus–Malus systems). For instance: Several mathematical tools for pricing a rate–making system has been provided by Lange (1969). Dionne & Vanasse (1989, 1992) employed available asymmetric information under Poisson and Negative Binomial regression models to determine premium of a rate–making system. In 1995, Lemaire designed an optimal Bonus–Malus system based on Negative Binomial distribution. Pinquet (1997) considered Poisson and Lognormal distributions to design an optimal Bonus–Malus system. Walhin & Paris (1999) considered a Hofmann’s distribution along with a finite mixture Poisson distribution to evaluate elements of a Bonus–Malus system. The relatively premium of a rate–making system under the exponential loss function has been evaluated by Denuit & Dhaene (2001). In 2001, Frangos & Vrontos designed an optimal Bonus–Malus system using both Pareto and Negative Binomial distributions. Using the bivariate Poisson regression model Bermúdez & Morata (2009) studied priori rate–making procedure for an automobile insurance database which has two different types of claims. In 2011, Bermúdez & Karlis employed a Bayesian multivariate Poisson model to determine premium of a rate–making system which has a non-ignorable correlation between types of its claims. Boucher & Inoussa (2014) introduced a new model to determine premium of a rate–making system whenever panel or longitudinal data are available. The Sichel distribution along with a Negative Binomial distribution have been considered by Tzougas & Frangos (2013, 2014a, 2014b). Tzougas et al. (2014) employed a finite mixture distribution to model frequency and severity of accidents. Payandeh Najafabadi et al. (2015) employed Payandeh Najafabadi (2010)’s idea to determine credibility premium for a rate–making system whenever number of reported claims distributed according to a zero-inflated Poisson distribution. Several authors have been employed zero-inflated models in actuarial science, see instance Yip & Yau (2005), Boucher et al. (2009), Boucher & Denuit (2008), and Boucher et al. (2007), among others.

Under a rate–making system the pure premium of an $i^{\hbox{th}}$ policyholder at $(t+1)^{\hbox{th}}$ year has been estimated by multiplication of estimated base premium, say ${\hat{B}P}(t+1),$ into corresponding estimated rate premium, say ${\hat{R}ate}(t+1).$ From decision theory point of view, the Bayes estimator offers an intellectually and acceptable estimation for both the rate premium $Rate(t+1)$ and the base premium $BP(t+1).$ Such Bayes estimators, under the quadratic loss function, can be obtained by posterior expectation of risk parameters given number and severity of reported claims at first $t+1$ years, see Denuit et al. (2007) for more details.

Therefore, to determine premium for $i^{\hbox{th}}$ policyholder, under a rate–making system, one has to determine both Bayes estimators. The following two theorems develop such estimators. Namely, in the first step, it supposes that number of reported claim $Y_{1},\cdots,Y_{t},$ given risk parameter $\Lambda_{i}=\lambda_{i},$ has been distributed according to a k-Inflated Poisson distribution and risk parameter $\Lambda_{i}$ distributed as a finite mixture Gamma. In the second step, it supposes that claim size random variable $Z_{1},\cdots,Z_{t},$ given risk parameter $\Theta_{i}=\theta_{i},$ has been distributed according to an exponential distribution and risk parameter $\Theta_{i}$ distributed as a finite mixture inverse Gamma. Finally, it derives such Bayes estimators for risk parameters $\theta_{i}$ and $\lambda_{i}.$

Theorem 1**.**

*Suppose that for an $i^{\hbox{th}}$ policyholder, number of reported claims in the last $t$ years have been restated as ${\bf Y}_{i}=(Y_{i1},\cdots,Y_{it}).$ Also suppose that, for $l=1,\cdots,t,$ $Y_{il}$ given parameter $\Lambda_{il}=\lambda_{ik}$ has been distributed according to a k-Inflated Poisson distribution with probability mass function $P(Y_{il}=y_{il}|\lambda_{il})=pI_{\{k\}}(y_{il})+q\exp(-\lambda_{il})(\lambda_{il})^{y}_{il}/y_{il}!,$ where $p~{}\&~{}q\in[0,1]$ and $p+q=1.$ Moreover, suppose that risk parameter $\Lambda_{i}$ can be restated as regression model $\log({\lambda}_{il})=C_{i}B_{il}+{\epsilon}_{i}$ where ${\bf C}_{i}=(1,c_{i,1},\dots,c_{i,p})$ is the vector of $p$ characteristics/covariates for an $i^{\hbox{th}}$ policyholder, $B_{il}=(\beta_{0il},\cdots,\beta_{pil})^{\prime}$ is the vector of the regression coefficients, and $u_{i}=\exp(\epsilon_{i})$ has been distributed according to finite mixture gamma distribution with density function $f_{U_{i}}(u_{i})=\sum^{m}_{j=1}{{\varphi}_{j}{{u_{i}}^{{\alpha}_{j}-1}{{\alpha}_{j}}^{{\alpha}_{j}}e^{-{\alpha}_{j}u_{i}}}/{\mathit{\Gamma}({\alpha}_{j})}},$ where $u_{i}>0$, ${\alpha}_{j}>0$ and $\sum^{m}_{j=1}{{\varphi}_{j}=1}$. Then, Bayes estimator for the rate premium $\widehat{Rate}_{i}(t+1),$ of an $i^{\hbox{th}}$ policyholder at $(t+1)^{\hbox{th}}$ year, is given by *

[TABLE]

where $k_{j}(u_{i}):={{\varphi}_{j}{{u_{i}}^{{\alpha}_{j}-1}{{\alpha}_{j}}^{{\alpha}_{j}}e^{-{\alpha}_{j}u_{i}}}/{\mathit{\Gamma}({\alpha}_{j})}},$ ${\mathit{\Gamma}(\cdot)}$ stands for the Gamma function, and $h_{il}(u_{i}):=pI_{\left(y_{il}=k\right)}+qe^{-\exp(C_{i}B_{il})u_{i}}{\left(\exp(C_{i}B_{il})u_{i}\right)}^{y_{il}}/y_{il}!.$

Proof. The Bayes estimator for the rate premium $Rate_{i}(t+1),$ under the quadratic loss function, is mean of posterior distribution $\Lambda_{it+1}|({\bf Y}_{i},{\bf C}_{i}).$ Such the posterior distribution can be restated as the following.

[TABLE]

Now the desired result arrives by

[TABLE]

In a situation that $q=1,$ the rate premium ${\hat{R}ate}_{i}(t+1)$ can be restated as

[TABLE]

where $y_{i\cdot}=\sum_{l=1}^{t}y_{il}.$ This situation has been studied by Dionne & Vanasse (1992) for an one mixture distribution and by Tzougas et al. (2014) for an $m$ mixture distribution. In the case that $t=0,$ one may show that $\widehat{Rate}_{i}(1)=e^{C_{i}B_{i1}}.$

Remark 1**.**

*For the situation that no covariate information has been taken into account, say a distribution model, and the risk parameter $\Lambda_{i}$ has been distributed according to a finite mixture gamma distribution with density function given by (3). Result of Theorem (1) can be reformulated as *

[TABLE]

The following theorem develops a Bayes estimator for the base premium $BP_{i}(t+1)$ for an $i^{\hbox{th}}$ policyholder at $(t+1)^{\hbox{th}}$ year.

Theorem 2**.**

Suppose that for an $i^{\hbox{th}}$ policyholder, severity/size of claims in the last $t$ years have been restated as ${\bf Z}_{i}=({\bf Z}_{i1},\cdots,{\bf Z}_{it}).$ Also suppose that, for $l=1,\cdots,t,$ ${\bf Z}_{il}=(Z_{il1},\cdots,Z_{ilk_{il}}),$ where $k_{il}$ stands for number of reported claims by $i^{\hbox{th}}$ policyholder at $l^{\hbox{th}}$ year, and for $s=1,\cdots,k_{il},$ assume that $Z_{ils}$ given parameter $\Theta_{il}=\theta_{il}$ has been distributed according to an exponential distribution function with density function $f_{Z_{il}|\Theta_{il}=\theta_{il}}(z_{il})=\exp\{-z_{il}/\theta_{il}\}/\theta_{il}.$ Moreover, suppose that risk parameter $\Theta_{i}$ can be restated as $\log({\theta}_{i,l})={\bf W}_{i}{\bf D}_{il}+\epsilon_{i},$ where ${\bf W}_{i}=(1,w_{i,1},\dots,w_{i,p})$ is the vector of $p$ characteristics/covariates for an $i^{\hbox{th}}$ policyholder, ${\bf D}_{il}=(d_{0il},\cdots,d_{pil})^{\prime}$ is the vector of the regression coefficients, and $u_{i}=\exp(\epsilon_{i})$ has been distributed according to a finite mixture Inverse Gamma with density function $f_{U_{i}}(u_{i})=\sum_{j=1}^{m}\phi_{j}(\eta_{j}-1)^{\eta_{j}}u_{i}^{-\eta_{j}-1}\exp(-(\eta_{j}-1)/u_{i})/\Gamma(\eta_{j}),$ where $u_{i}>0$, ${\eta}_{j}>0$ and $\sum^{m}_{j=1}{\phi_{j}=1}$. Then, Bayes estimator for the the base premium $BP_{i}(t+1)$ for an $i^{\hbox{th}}$ policyholder at $(t+1)^{\hbox{th}}$ year, is given by

[TABLE]

where $K_{i}=\sum_{l=1}^{t}k_{il}.$

Proof. The posterior distribution of $\Theta_{it+1}|({\bf Z}_{i},{\bf W}_{i})$ can be restated as

[TABLE]

The desired Bayes estimator arrives by

[TABLE]

The above result also obtained by Tzougas et al. (2014).

Remark 2**.**

For the situation that no covariate information has been taken into account, say a distribution model, and the risk parameter $\Theta_{i}$ has been distributed according to a finite mixture Inverse Gamma with density function given by (7). Result of Theorem (2) can be reformulated as

[TABLE]

To show practical application of our findings, the next section provides an real example.

5 Numerical Application

Now, we considered available data from Iranian third party liability, at 2011 year. After a primary investigation, we just trusted information about 8874 policyholders. We used 4 independent variables, as covariates, presented in Table 1. For each policyholder we have the initial information at the beginning of the period and we are interested such covariates to model frequency/severity of claims for evaluating pure premium of each policyholder under a rate–making system.

Table 1: Available covariates information for each policyholder.

Variable Description

Gender Equal to 0 for woman & 1 for man

Age Equal to 1 for $18\leq age<30;$ 2 for $30\leq age<40;$ 3 for $40\leq age<50;$ & 4 for $50\leq age$

Car’s price Equal to 1 for $price<2\times 10^{4};$ 2 for $2\times 10^{4}\leq price<5\times 10^{4};$ 3 for $5\times 10^{4}\leq price<10^{5};$ & 4 for $10^{5}\leq price$

Living area Equal to 1 for $population~{}size<10^{5};$ 2 for $10^{5}\leq population~{}size<5\times 10^{5};$ 3 for $5\times 10^{5}\leq population~{}size<10^{6};$

& 4 for $10^{6}\leq population~{}size$

For simplicity in presentation hereafter, we represent $kINBM_{m}$ for a k-Inflated Negative Binomial model with $m$ mixture components and Pareto$M_{m}$ for a Pareto model with $m$ mixture components.

To find an appropriate distribution for the frequency of claim, in the first step, we considered the $kINBM_{m}$ model along with all distributions that have been considered, by authors, to model frequency of claims in a rate–making system. Namely, we considered the kINBM, Delaporte, Sichel, and Poisson Inverse Gaussian, say PIG, distributions for frequency and the Pareto$M_{m}$ distribution for severity and estimate their parameters.

The maximum likelihood estimator for the $kINBM_{m}$ we develop our R codes while the maximum likelihood estimator for other distributions have been computed using the GAMLSS package in R. Table 2 represents the maximum likelihood estimator for significant parameters of such distributions. The significant test for each parameter has been tested by the Wald test.

Now using a backward elimination selection method, we find covariates that may impact on response variable for each regression model. The significant test for each covariate has been done by the Wald test. Table 3 shows result of the backward selection method for frequency/severity of accidents.

Table 2. Estimation for parameters on various model for frequency/severity of claims.

Distribution:

$NBM_{1}$ $0INBM_{1}$ $1INBM_{1}$ $2INBM_{1}$ $3INBM_{1}$ $NBM_{2}$ $0INBM_{2}$

${\boldsymbol{\omega}}=(*,1)$ ${\boldsymbol{\omega}}=(0.001,0.999)$ ${\boldsymbol{\omega}}=(0.136,0.861)$ ${\boldsymbol{\omega}}=(0,1)$ ${\boldsymbol{\omega}}=(0.001,0.999)$ ${\boldsymbol{\omega}}=(*,0.005,0.995)$ ${\boldsymbol{\omega}}=(0.001,0.004,0.995)$

${\boldsymbol{\tau}}=23.390$ ${\boldsymbol{\tau}}=22.256$ ${\boldsymbol{\tau}}=1.755$ ${\boldsymbol{\tau}}=19.408$ ${\boldsymbol{\tau}}=32.333$ ${\boldsymbol{\tau}}=(14.152,141.857)$ ${\boldsymbol{\tau}}=(26.027,124.000)$

${\boldsymbol{\alpha}}=5.717$ ${\boldsymbol{\alpha}}=5.376$ ${\boldsymbol{\alpha}}=0.217$ ${\boldsymbol{\alpha}}=4.734$ ${\boldsymbol{\alpha}}=7.730$ ${\boldsymbol{\alpha}}=(39.02,32.53)$ ${\boldsymbol{\alpha}}=(71.74,30.31)$

Distribution:

$1INBM_{2}$ $2INBM_{2}$ $3INBM_{2}$ $NBM_{3}$ $0INBM_{3}$ $1INBM_{3}$ Delaporte

${\boldsymbol{\omega}}=(0.116,0.831,0.053)$ ${\boldsymbol{\omega}}=(0,1,0)$ ${\boldsymbol{\omega}}=(0.001,0.997,0.002)$ ${\boldsymbol{\omega}}=(*,0.014,0.982,0.004)$ ${\boldsymbol{\omega}}=(0.003,0.007,0.986,0.004)$ ${\boldsymbol{\omega}}=(0.125,0.644,0.033,0.198)$ $\lambda=0.243$

${\boldsymbol{\tau}}=(8.174,2.690)$ ${\boldsymbol{\tau}}=(19.408,30.250)$ ${\boldsymbol{\tau}}=(25.316,42.478)$ ${\boldsymbol{\tau}}=(124.000,124.000,30.250)$ ${\boldsymbol{\tau}}=(141.857,141.857,27.571)$ ${\boldsymbol{\tau}}=(4.587,2.623,4.181)$ $\sigma=77.67$

${\boldsymbol{\alpha}}=(0.807,2.305)$ ${\boldsymbol{\alpha}}=(4.735,6.327)$ ${\boldsymbol{\alpha}}=(6.054,8.88)$ ${\boldsymbol{\alpha}}=(30.47,29.48,85.17)$ ${\boldsymbol{\alpha}}=(33.70,32.58,77.67)$ ${\boldsymbol{\alpha}}=(0.357,2.628,0.729)$ $\nu=0.913$

Distribution:

Sichel PIG Pareto$M_{1}$ Pareto$M_{2}$ Pareto$M_{3}$

$\mu=0.242$ $\mu=0.242$ ${\boldsymbol{\rho}}=1$ ${\boldsymbol{\rho}}=(0.519,0.481)$ ${\boldsymbol{\rho}}=(0.332,0.321,0.347)$

$\sigma=NS$ $\sigma=0.225$ ${\boldsymbol{\alpha}}=1.871$ ${\boldsymbol{\alpha}}=(1.871,1.871)$ ${\boldsymbol{\alpha}}=(1.871,1.873,1.873)$

$\nu=-4.961$ — ${\boldsymbol{\gamma}}=16.44$ ${\boldsymbol{\gamma}}=(16.43,16.44)$ ${\boldsymbol{\gamma}}=(16.44,16.43,16.43)$

where the first element in ${\boldsymbol{\omega}}$ stands for weight of inflated part and we use $*$ whenever the distribution is non-inflated distribution and $NS$ stands for not significant at 5% level.

Table 3. Regression coefficients for various model for frequency/severity of claims.

Regression model:

$NBM_{1}$ $0INBM_{1}$ $1INBM_{1}$ $2INBM_{1}$ $3INBM_{1}$ $NBM_{2}$ $0INBM_{2}$

${\boldsymbol{\omega}}=(*,1)$ ${\boldsymbol{\omega}}=(0.041,0.959)$ ${\boldsymbol{\omega}}=(0.111,0.889)$ ${\boldsymbol{\omega}}=(0.001,0.999)$ ${\boldsymbol{\omega}}=(0.002,0.998)$ ${\boldsymbol{\omega}}=(*,0.893,0.107)$ ${\boldsymbol{\omega}}=(0.047,0.025,0.928)$

${\boldsymbol{\alpha}}=7.560$ ${\boldsymbol{\alpha}}=22.56$ ${\boldsymbol{\alpha}}=0.440$ ${\boldsymbol{\alpha}}=7.950$ ${\boldsymbol{\alpha}}=9.980$ ${\boldsymbol{\alpha}}=(0.074,22.01)$ ${\boldsymbol{\alpha}}=(0.061,18.71)$

Intercept ${\boldsymbol{\beta}_{0}}=-0.738$ ${\boldsymbol{\beta}_{0}}=-0.698$ ${\boldsymbol{\beta}_{0}}=-0.887$ ${\boldsymbol{\beta}_{0}}=-0.744$ ${\boldsymbol{\beta}_{0}}=-0.745$ ${\boldsymbol{\beta}_{0}}=(NS,-0.766)$ ${\boldsymbol{\beta}_{0}}=(NS,-0.719)$

Gender ${\boldsymbol{\beta}_{1}}=NS$ ${\boldsymbol{\beta}_{1}}=NS$ ${\boldsymbol{\beta}_{1}}=NS$ ${\boldsymbol{\beta}_{1}}=NS$ ${\boldsymbol{\beta}_{1}}=NS$ ${\boldsymbol{\beta}_{1}}=(NS,NS)$ ${\boldsymbol{\beta}_{1}}=(NS,NS)$

Age ${\boldsymbol{\beta}_{2}}=-0.118$ ${\boldsymbol{\beta}_{2}}=-0.118$ ${\boldsymbol{\beta}_{2}}=-0.213$ ${\boldsymbol{\beta}_{2}}=-0.121$ ${\boldsymbol{\beta}_{2}}=-0.114$ ${\boldsymbol{\beta}_{2}}=(NS,-0.116)$ ${\boldsymbol{\beta}_{2}}=(NS,-0.117)$

Car’s price ${\boldsymbol{\beta}_{3}}=-0.189$ ${\boldsymbol{\beta}_{3}}=-0.189$ ${\boldsymbol{\beta}_{3}}=-0.263$ ${\boldsymbol{\beta}_{3}}=-0.190$ ${\boldsymbol{\beta}_{3}}=-0.196$ ${\boldsymbol{\beta}_{3}}=(0.709,0.175)$ ${\boldsymbol{\beta}_{3}}=(0.587,0.180)$

Living area ${\boldsymbol{\beta}_{4}}=NS$ ${\boldsymbol{\beta}_{4}}=NS$ ${\boldsymbol{\beta}_{4}}=NS$ ${\boldsymbol{\beta}_{4}}=NS$ ${\boldsymbol{\beta}_{4}}=NS$ ${\boldsymbol{\beta}_{4}}=(NS,NS)$ ${\boldsymbol{\beta}_{4}}=(NS,NS)$

Regression model:

$1INBM_{2}$ $2INBM_{2}$ $3INBM_{2}$ $BM_{3}$ $0INBM_{3}$ $1INBM_{3}$ Delaporte

${\boldsymbol{\omega}}=(0.120,0.833,0.047)$ ${\boldsymbol{\omega}}=(0.002,0.024,0.974)$ ${\boldsymbol{\omega}}=(0.002,0.022,0.976)$ ${\boldsymbol{\omega}}=(*,0.93,0.02,0.05)$ ${\boldsymbol{\omega}}=(0.03,0.07,0.46,0.44)$ ${\boldsymbol{\omega}}=(0.13,0.07,0.40,0.40)$ $\sigma=59.56$

${\boldsymbol{\alpha}}=(0.685,9.471)$ ${\boldsymbol{\alpha}}=(0.087,19.00)$ ${\boldsymbol{\alpha}}=(0.090,18.94)$ ${\boldsymbol{\alpha}}=(31.22,30.95,0.030)$ ${\boldsymbol{\alpha}}=(13.29,11.32,0.157)$ ${\boldsymbol{\alpha}}=(9.562,0.643,0.396)$ $nu=0.910$

Intercept ${\boldsymbol{\beta}_{0}}=(1.009,0.663)$ ${\boldsymbol{\beta}_{0}}=(NS,-0.775)$ ${\boldsymbol{\beta}_{0}}=(NS,-0.775)$ ${\boldsymbol{\beta}_{0}}=(.0.638,1.032,NS)$ ${\boldsymbol{\beta}_{0}}=(0.531,0.384,0.165)$ ${\boldsymbol{\beta}_{0}}=(0.535,2.514,2.754)$ ${\boldsymbol{\beta}_{0}}=-0.749$

Gender ${\boldsymbol{\beta}_{1}}=(-0.185,NS)$ ${\boldsymbol{\beta}_{1}}=(NS,NS)$ ${\boldsymbol{\beta}_{1}}=(NS,NS)$ ${\boldsymbol{\beta}_{1}}=(NS,NS,NS)$ ${\boldsymbol{\beta}_{1}}=(NS,NS,NS)$ ${\boldsymbol{\beta}_{1}}=(0.770,0.632,NS)$ ${\boldsymbol{\beta}_{1}}=NS$

Age ${\boldsymbol{\beta}_{2}}=(NS,-0.442)$ ${\boldsymbol{\beta}_{2}}=(NS,-0.115)$ ${\boldsymbol{\beta}_{2}}=(NS,-0.114)$ ${\boldsymbol{\beta}_{2}}=(0.305,0.337,NS)$ ${\boldsymbol{\beta}_{2}}=(NS,0.151,0.165)$ ${\boldsymbol{\beta}_{2}}=(0.343,0.548,0.813)$ ${\boldsymbol{\beta}_{2}}=-0.113$

Car’s price ${\boldsymbol{\beta}_{3}}=(0.607,0.076)$ ${\boldsymbol{\beta}_{3}}=(0.618,0.177)$ ${\boldsymbol{\beta}_{3}}=(0.675,0.179)$ ${\boldsymbol{\beta}_{3}}=(NS,0.676,0.332)$ ${\boldsymbol{\beta}_{3}}=(0.230,0.151,0.310)$ ${\boldsymbol{\beta}_{3}}=(NS,0.298,1.545)$ ${\boldsymbol{\beta}_{3}}=-0.189$

Living area ${\boldsymbol{\beta}_{4}}=(NS,NS)$ ${\boldsymbol{\beta}_{4}}=(NS,NS)$ ${\boldsymbol{\beta}_{4}}=(NS,NS)$ ${\boldsymbol{\beta}_{4}}=(NS,NS,NS)$ ${\boldsymbol{\beta}_{4}}=(NS,NS,NS)$ ${\boldsymbol{\beta}_{4}}=(0.238,2.382,0.623)$ ${\boldsymbol{\beta}_{4}}=NS$

Regression model:

Sichel PIG Pareto$M_{1}$ Pareto$M_{2}$ Pareto$M_{3}$

$\sigma=NS$ $\sigma=0.174$ ${\boldsymbol{\rho}}=1$ ${\boldsymbol{\rho}}=(0.542,0.458)$ ${\boldsymbol{\rho}}=(0.341,0.312,0.347)$

$\nu=-5.688$ — ${\boldsymbol{\alpha}}=1.927$ ${\boldsymbol{\alpha}}=(1.927,1.927)$ ${\boldsymbol{\alpha}}=(1.93,1.93,1.93)$

Intercept ${\boldsymbol{\beta}_{0}}=-0.737$ ${\boldsymbol{\beta}_{0}}=-0.736$ ${\boldsymbol{\beta}_{0}}=16.15$ ${\boldsymbol{\beta}_{0}}=(6.15,6.15)$ ${\boldsymbol{\beta}_{0}}=(16.15,16.15,16.15)$

Gender ${\boldsymbol{\beta}_{1}}=NS$ ${\boldsymbol{\beta}_{1}}=NS$ ${\boldsymbol{\beta}_{1}}=NS$ ${\boldsymbol{\beta}_{1}}=(NS,NS)$ ${\boldsymbol{\beta}_{1}}=(NS,NS,NS)$

Age ${\boldsymbol{\beta}_{2}}=-0.119$ ${\boldsymbol{\beta}_{2}}=-0.119$ ${\boldsymbol{\beta}_{2}}=NS$ ${\boldsymbol{\beta}_{2}}=(NS,NS)$ ${\boldsymbol{\beta}_{2}}=(NS,NS,NS)$

Car’s price ${\boldsymbol{\beta}_{3}}=-0.190$ ${\boldsymbol{\beta}_{3}}=-0.190$ ${\boldsymbol{\beta}_{3}}=0.157$ ${\boldsymbol{\beta}_{3}}=(-0.16,-0.16)$ ${\boldsymbol{\beta}_{3}}=(-0.16,-0.16,-0.16)$

Living area ${\boldsymbol{\beta}_{4}}=NS$ ${\boldsymbol{\beta}_{4}}=NS$ ${\boldsymbol{\beta}_{4}}=NS$ ${\boldsymbol{\beta}_{4}}=(NS,NS)$ ${\boldsymbol{\beta}_{4}}=(NS,NS,NS)$

where the first element in ${\boldsymbol{\omega}}$ stands for weight of inflated part and we use $*$ whenever the distribution is non-inflated distribution and $NS$ stands for not significant at 5% level.

5.1 Model comparison

To obtain an appropriate model for a given rate–making system, this section begins by considering the $kINBM_{m}$ model along with all distributions that have been considered, by authors, to model frequency of claims in a rate–making system. Now in order to compare result of regression/distribution models, we conducted three evaluation approaches. Namely: (1) In the first approach, to study performance of count distributions, we employ each fitted distribution, 200 times, to simulate 8874 data. Then, using the mean square error, say MSE, criteria, we compare stimulated data with observed data (see Table 3 for result on such comparison); (2) The second approach provides a pairwise comparison between fitted count regression/distribution models based upon *either * the Vuong test (for two non-nested models) or the likelihood ratio test (for two nested models), see Table 4 for such comparison study; and finally, (3) The third approach employs the Akaike Information Criterion (AIC) and the Schwarz Bayesian information Criterion (SBIC) to compare regression/distribution models for both frequency and severity of claims, result of such comparison has been reported in Table 5.

Generating Data approach:

To study performance of fitted count distributions given in Table

1. We employ the GAMLSS package in R to generate samples from the Delaporte, the Sichel, the Poisson Inverse Gaussian distributions. Lim et al. (2014) introduced an idea to generate sample from a given Zero-inflated Poisson mixture distribution. Now, we employ their idea to generate samples from a given a $kINBM_{m}$ distribution. Based upon their idea, to generate sample $y_{i}$ from a $kINBM_{m}$ distribution with probability mass function

[TABLE]

where all parameters ${\boldsymbol{\theta}}=({\boldsymbol{\alpha}},{\boldsymbol{\tau}},{\boldsymbol{p}})$ and $k$ are given. We start with a dummy variable, say $s_{i},$ which generated from an uniform (0,1) distribution. If $0\leq s_{i}\leq p_{1},$ we set $y_{i}=k;$ If $p_{1}<s_{i}\leq p_{1}+p_{2},$ then $y_{i}$ is a draw from a Negative Binomial distribution $(\alpha_{1},\alpha_{1}/(\alpha_{1}+\tau_{1}));$ If $p_{1}+p_{2}<s_{i}\leq p_{1}+p_{2}+p_{3},$ then $y_{i}$ is a draw from a Negative Binomial distribution $(\alpha_{2},\alpha_{2}/(\alpha_{2}+\tau_{2}));$ If $p_{1}+p_{2}+p_{3}<s_{i}\leq p_{1}+p_{2}+p_{3}+p_{4},$ then $y_{i}$ is a draw from a Negative Binomial distribution $(\alpha_{3},\alpha_{3}/(\alpha_{3}+\tau_{3}));$ and so on.

We employed the GAMLSS package and the above idea to simulate 8874 data (200 times). Table 4 reports mean (mean square error, say MSE) of frequency for such 200 times simulated samples.

Table 4. Mean and the MSE of frequency for generated data under count distributions given in Table 1.

Mean (MSE) of frequency for generated data under Distribution:

Observed(Freq.) Delaporte Sichel PIG $NBM_{1}$ $0INBM_{1}$ $1INBM_{1}$

0(6956) 7018.240(5134.78) 7027.435(6384.40) 7001.370(4548.36) 69980.515(1579.14) 7001.825(2964.21) 6958.575(1498.88)

1(1751) 1614.615(19105.90) 1584.62(28655.82) 1620.145(22066.28) 1626.805(13343.36) 1625.580(16799.97) 1748.300(1598.76)

2(122) 203.210(6774.82) 227.235(11188.30) 224.160(11090.70) 222.960(11472.20) 221.325(11189.41) 120.835(136.98)

3(31) 25.540(55.10) 29.680(26.90) 25.585(49.58) 23.645(81.46) 22.950(73.84) 32.555(33.04)

4(9) 6.470(12.50) 4.230(24.28) 2.485(39.78) 1.910(48.10) 2.145(44.23) 9.510(9.50)

5(3) 2.940(1.82) 0.595(6.12) 0.230(7.78) 0.175(6.94) 0.175(7.90) 2.910(2.54)

6(2) 1.430(1.36) 0.160(3.74) 0.000(4.00) 0.000(4.00) 0.00(4.00) 0.895(1.54)

$>6$(0) 1.560(2.88) 0.000(0.00) 0.000(0.00) 0.000(0.00) 0.00(0.00) 0.420(0.53)

Mean (MSE) of frequency for generated data under Distribution:

Observed(Freq.) $2INBM_{1}$ $3INBM_{1}$ $NBM_{2}$ $0INBM_{2}$ $1INBM_{2}$ $2INBM_{2}$

0(6956) 6988.835(2476.51) 7003.555(3652.21) 7019.265(6530.50) 7019.135(3004.08) 6958.730(1529.84) 7002.304(3033.16)

1(1751) 1624.415(15079.51) 1626.055(16892.21) 1619.010(22792.00) 1616.830(6657.27) 1748.690(1448.56) 1621.425(16415.15)

2(122) 232.030(11339.35) 214.305(8663.16) 198.745(6347.56) 200.715(10290.01) 122.380(124.82) 220.400(10608.91)

3(31) 26.405(72.11) 28.205(20.45) 23.780(46.54) 23.940(35.47) 30.530(24.16) 26.855(53.16)

4(9) 2.015(41.56) 1.750(51.70) 7.010(11.76) 7.150(10.84) 9.030(8.68) 2.800(38.85)

5(3) 2.910(2.54) 0.155(8.51) 3.635(2.79) 3.350(3.60) 3.115(2.56) 0.155(9.01)

6(2) 0.115(3.85) 0.210(3.38) 1.730(1.18) 1.640(2.04) 1.075(1.26) 0.095(3.175)

$>6$(0) 0.000(0.00) 0.000(0.00) 1.11(3.44) 1.090(2.13) 0.450(0.60) 0.000(0.000)

Mean (MSE) of frequency for generated data under Distribution:

Observed(Freq.) $3INBM_{2}$ $BM_{3}$ $0INBM_{3}$ $1INBM_{3}$

0(6956) 7019.200(5955.36) 7015.995(1585.00) 7019.785(7553.77) 6959.035(1127.89)

1(1751) 1605.105(22133.02) 1619.510(11018.88) 1617.120(21482.31) 1747.870(1038.81)

2(122) 215.755(7985.44) 200.635(8149.00) 199.745(5686.31) 121.985(213.64)

3(31) 32.100(56.55) 24.635(65.20) 24.355(124.61) 30.630(52.47)

4(9) 1.650(54.61) 6.9100(10.76) 6.955(12.64) 9.605(8.34)

5(3) 0.205(9.04) 3.635(2.78) 3.400(3.20) 3.245(1.94)

6(2) 0.000(4.00) 1.600(1.14) 1.595(0.675) 1.215(1.37)

$>6$(0) 0.000(0.00) 0.650(1.28) 1.045(2.01) 0.415(0.68)

Results of the simulation study, given in Table 4, shows that the MSE for the all 1-Inflated Negative Binomial mixture distributions, is considerably less than the MSE of other fitted distributions. Therefore, based upon this simulation study, one may conclude that the 1-Inflated Negative Binomial mixture distributions are appropriate distributions for claim frequency of Iranian policyholders.

The Vuong and the likelihood ratio tests’ approach:

To make a decision about statistical hypothesis

[TABLE]

If both of distributions are belong to a family of distributions with different parameters (nested models), one may employ the likelihood ratio test to make such decision. Otherwise, where models are belong to two different family of distributions (Non-nested models) the Vuong has to used, see Denuit et al. (2007, §2) for more details.

Table 5 represents a pairwise comparison between fitted count regression/distribution models given in Tables 1 and 2.

Table 5. Result of the Vuong test (for two non-nested models) or the likelihood ratio test (for two nested models).

Panel A: Result of the Vuong test

Model 1 Model 2 Decision on fitted regression Decision on fitted distribution

Delaporte $1INBM_{1}$ $1INBM_{1}$ (Statistic=-37.63 & P-value=0.00) $1INBM_{1}$ (Statistic=-51.58 & P-value=0.00)

PIG $1INBM_{1}$ $1INBM_{1}$ (Statistic=-33.15 & P-value=0.00) $1INBM_{1}$ (Statistic=-38.94 & P-value=0.00)

$0INBM_{1}$ $1INBM_{1}$ $1INBM_{1}$ (Statistic=-24.75 & P-value=0.00) $1INBM_{1}$ (Statistic=-35.57 & P-value=0.00)

$2INBM_{1}$ $1INBM_{1}$ $1INBM_{1}$ (Statistic=-26.80 & P-value=0.00) $1INBM_{1}$ (Statistic=-37.09 & P-value=0.00)

$3INBM_{1}$ $1INBM_{1}$ $1INBM_{1}$ (Statistic=-22.41 & P-value=0.00) $1INBM_{1}$ (Statistic=-31.22 & P-value=0.00)

$0INBM_{2}$ $1INBM_{2}$ $1INBM_{2}$ (Statistic=-43.51 & P-value=0.00) $1INBM_{2}$ (Statistic=-52.27 & P-value=0.00)

$2INBM_{2}$ $1INBM_{1}$ $1INBM_{2}$ (Statistic=-40.89 & P-value=0.00) $1INBM_{2}$ (Statistic=-37.22 & P-value=0.00)

$3INBM_{2}$ $1INBM_{2}$ $1INBM_{2}$ (Statistic=-40.36 & P-value=0.00) $1INBM_{2}$ (Statistic=-33.16 & P-value=0.00)

$0INBM_{3}$ $1INBM_{3}$ $1INBM_{3}$ (Statistic=-34.28 & P-value=0.00) $1INBM_{3}$ (Statistic=-20.48 & P-value=0.00)

Panel B: Result of the likelihood ratio test

Model 1 Model 2 Decision on fitted regression Decision on fitted distribution

$NBM_{1}$ $1INBM_{1}$ $1INBM_{1}$ (Statistic=62.21 & P-value=0.00) $1INBM_{1}$ (Statistic=105.00 & P-value=0.00)

$NBM_{2}$ $1INBM_{2}$ $1INBM_{2}$ (Statistic=80.46 & P-value=0.00) $1INBM_{2}$ (Statistic=49.66 & P-value=0.00)

$NBM_{3}$ $1INBM_{3}$ $1INBM_{3}$ (Statistic=111.65 & P-value=0.00) $1INBM_{3}$ (Statistic=49.75 & P-value=0.00)

$1INBM_{1}$ $1INBM_{2}$ $1INBM_{2}$ (Statistic=45.35 & P-value=0.00) $1INBM_{1}$ (Statistic=0.18 & P-value=0.90)

$1INBM_{1}$ $1INBM_{3}$ $1INBM_{2}$ (Statistic=88.65 & P-value=0.00) $1INBM_{1}$ (Statistic=0.21 & P-value=0.87)

$1INBM_{2}$ $1INBM_{3}$ $1INBM_{3}$ (Statistic=42.31 & P-value=0.00) $1INBM_{2}$ (Statistic=0.3 & P-value=0.97)

Based upon results of Table 5, one may conclude that the 1-Inflated Negative Binomial mixture distributions/regressions, at 5% significant level, defeat other distributions/regressions.

The Akaike Information Criterion (AIC) and the

Schwarz Bayesian information criterion approaches:

The Akaike Information Criterion (AIC) and the Schwarz Bayesian Information Criterion (SBIC) are two measure to select an appropriate model among a set of candidate models. Both criteria are defined based on -2 times the maximum log-likelihood, penalized by either number of estimated parameters, for AIC, or number of estimated parameters times logarithm of number of observations, for SBIC. Given a set of candidate models, a preferred model is the one which has the minimum AIC (SBIC) value, see Denuit et al. (2007, §1) for more details.

Table 6 provides the AIC and the SBIC for fitted regression/distribution models for both frequency and severity of claims.

Table 6. Result of the Akaike Information Criterion (AIC) and the Schwarz Bayesian information Criterion (SBIC).

Regression model

Distribution model

Model df AIC SBIC df AIC SBIC

$NBM_{1}$ 6 10656.42 10698.88 2 10784.70 10798.88

Delaporte 7 10615.42 10665.07 3 10734.99 10756.26

Sichel 7 10648.96 10699.16 3 10772.67 10793.94

PIG 6 10653.90 10696.47 2 10781.11 10795.29

$0INBM_{1}$ 7 10664.59 10714.25 3 10786.74 10808.01

$1INBM_{1}$ 7 10596.11 10645.77 3 10681.69 10702.96

$2INBM_{1}$ 7 10658.36 10708.02 3 10787.05 10808.33

$3INBM_{1}$ 7 10653.08 10702.73 3 10783.25 10804.53

$INBM_{2}$ 13 10635.22 10727.86 5 10735.14 10770.59

$0INBM_{2}$ 14 1064.07 10746.80 6 10737.30 10779.84

$1INBM_{2}$ 14 10558.76 10658.58 6 10687.80 10730.02

$2INBM_{2}$ 14 10638.66 10748.39 6 10793.05 10835.60

$3INBM_{2}$ 14 10632.58 10732.31 6 10789.88 10832.42

$NBM_{3}$ 20 10632.11 10775.11 8 10741.26 10797.99

$0INBM_{3}$ 21 10654.22 10803.49 9 10743.23 10807.05

$1INBM_{3}$ 21 10536.45 10685.28 9 10693.19 10757.33

Pareto$M_{1}$ 6 63948.20 63990.75 2 64102.30 64116.48

Pareto$M_{2}$ 13 63968.20 64054.38 5 64108.30 64143.75

Pareto$M_{3}$ 20 63974.20 64108.93 8 64114.30 64171.00

The AIC and SBIC for fitted models, given Table 6, show that the 1-Inflated Negative Binomial mixture distributions/regressions are better than other distribution/regression models.

6 Rate–making Examples

To show practical application of our findings. We calculate the rate and pure premiums for the set of well fitted distributions/regression models that were presented in above sections. Since we are interested in the differences between rate premium of various classes. Therefore, we set the rate premium for a new policyholder equal to 1 unite, at $t=0.$ Moreover, we considered three different categories, described in Table 7.

Table 7: Categories which considered to evaluate rate and pure premiums under well fitted models.

Category Description

$A_{1}$ For a situation that no covariate information have been used for premium calculation

$A_{2}$ Whenever, chosen policyholder is a young man at age of 25 years old who owns a car

with price greater than $2\times 10^{4}$ and living in a city with population size larger than $10^{6}.$

$A_{3}$ Whenever, chosen policyholder is a mature woman at the age of 55 years old who owns

a car with price less than $2\times 10^{4}$ and living in a city with population size less than $10^{5}.$

Now to calculate rate premium for three categories $A_{1},$ $A_{2},$ and $A_{3},$ given in Table 7, using well fitted models. We consider two different approaches. The first approach just considers number of cumulated claims in the last yeas. While the second approach considers the exact number of reported claim for each year in a history of the policyholder. 222It worthwhile to mention that the second approach can be used just for inflated models.

Tables 8 and 9 represent calculated rate premium for three categories, given in Table 7, using well fitted models for both approaches.

Table 8: The rate premium for three categories $A_{1},$ $A_{2},$ and $A_{3}$ using well fitted models, whenever number of cumulated claims has been considered.

         Model:

Number of cumulated

$NBM_{1}$

$NBM_{2}$

$NBM_{3}$

Year claims up to this year ($K$) $A_{1}$ $A_{2}$ $A_{3}$ $A_{1}$ $A_{2}$ $A_{3}$ $A_{1}$ $A_{2}$ $A_{3}$

$t=0$ — 1 1 1 1 1 1 1 1 1

$K=0$ 0.96 0.96 0.98 0.95 0.91 0.98 0.95 0.87 0.90

$K=1$ 1.13 1.08 1.11 1.02 1.03 1.10 1.02 1.00 1.09

$t=1$ $K=2$ 1.29 1.21 1.24 1.54 1.18 1.35 1.42 1.13 1.96

$K=3$ 1.46 1.33 1.37 5.05 1.99 2.91 4.30 1.67 10.05

$K=4$ 1.63 1.46 1.50 10.40 5.78 9.12 9.76 4.88 24.99

$K=0$ 0.92 0.92 0.96 0.93 0.88 0.96 0.94 0.83 0.88

$K=1$ 1.08 1.04 1.09 0.97 1.00 1.08 0.98 0.97 1.04

$t=2$ $K=2$ 1.24 1.16 1.22 1.04 1.06 1.19 1.04 1.05 1.24

$K=3$ 1.40 1.28 1.35 1.53 1.18 1.66 1.40 1.15 2.39

$K=4$ 1.57 1.40 1.47 4.69 1.60 4.00 3.92 1.41 8.56

Table 9: The rate premium for three categories $A_{1},$ $A_{2},$ and $A_{3}$ using well fitted models, whenever exact number of reported claim for each year of the policyholder’s experience has been considered.

Model:

Year Number of reported

$1INBM_{1}$

$1INBM_{2}$

$1INBM_{3}$

claims at year $l$ ($k_{l}$) $A_{1}$ $A_{2}$ $A_{3}$ $A_{1}$ $A_{2}$ $A_{3}$ $A_{1}$ $A_{2}$ $A_{3}$

$t=0$ — 1 1 1 1 1 1 1 1 1

$k_{1}=0$ 0.64 0.63 0.88 0.83 0.81 0.96 0.79 0.63 0.96

$k_{1}=1$ 1.81 1.55 1.54 1.26 1.30 1.13 1.39 1.29 1.17

$t=1$ $k_{1}=2$ 6.52 3.91 4.86 2.50 2.43 2.55 3.55 2.54 2.21

$k_{1}=3$ 9.44 4.94 6.86 3.30 3.29 3.64 4.93 3.50 2.85

$k_{1}=4$ 12.37 6.38 8.85 4.13 4.12 4.54 6.31 4.46 3.49

$k_{1}=0,k_{2}=0$ 0.48 0.46 0.78 0.72 0.68 0.93 0.65 0.48 0.93

$k_{1}=0,k_{2}=1$ 1.15 1.03 1.33 1.06 1.05 1.09 1.10 0.84 1.03

$k_{1}=0,k_{2}=2$ 4.78 2.56 4.33 2.19 2.01 2.46 2.98 1.79 2.16

$k_{1}=1,k_{2}=0$ 1.15 1.03 1.33 1.06 1.05 1.09 1.10 0.84 1.13

$t=2$ $k_{1}=1,k_{2}=1$ 2.87 2.06 2.31 1.57 1.61 1.30 1.90 1.53 1.16

$k_{1}=1,k_{2}=2$ 6.79 3.58 5.63 2.76 2.60 2.82 3.93 2.47 2.38

$k_{1}=2,k_{2}=0$ 4.78 2.56 4.33 2.19 2.01 2.46 2.98 1.79 2.15

$k_{1}=2,k_{2}=1$ 6.79 3.58 5.63 2.76 2.60 2.82 3.93 2.47 2.38

$k_{1}=2,k_{2}=2$ 9.10 4.67 7.88 3.67 3.34 3.99 5.31 3.09 3.37

To illustrate a guideline to use result of Tables 8 and 9, suppose that either Negative Binomial with 2 mixture components, $NBM_{2},$ or 1-Inflated Negative Binomial with 2 mixture components, $1INBM_{2},$ can be considered as an appropriate model. Now consider the following three different scenarios.

Scenario 1:

For a given policyholder, no covariates information is available, category $A_{1}$ in Table 7. Based upon Table 8’s and Table 9’s result, respectively, his/her second year rate premium under $NBM_{2}$ model is 0.95 units while his/her second year rate premium under $1INBM_{2}$ model is 0.83 units, whenever such policyholder does not report any claim in the first year. But in the situation that such policyholder reports 2 claims in the first year. He/she has to pay 1.54 units, under $NBM_{2}$ model, and 2.50 units, under $1INBM_{2}$ model.

Scenario 2:

The given policyholder belongs to category $A_{2}$ of Table 7. Based upon Table 8’s and Table 9’s result, respectively, his second year rate premium, under $NBM_{2}$ model, is 0.91 units while his second year rate premium, under $1INBM_{2}$ model, is 0.81 units, whenever such policyholder does not report any claim in the first year. But in the situation that such policyholder reports 2 claims in the first year. He has to pay 1.18 units, under $NBM_{2}$ model, and 2.43 units, under $1INBM_{2}$ model.

Scenario 3:

The given policyholder belongs to category $A_{3}$ of Table 7. Based upon Table 8’s and Table 9’s result, respectively, her second year rate premium, under $NBM_{2}$ model, is 0.98 units while her second year rate premium, under $1INBM_{2}$ model, is 0.96 units, whenever such policyholder does not report any claim in the first year. But in the situation that such policyholder reports 2 claims in the first year. She has to pay 1.35 units, under $NBM_{2}$ model, and 2.55 units, under $1INBM_{2}$ model.

The above simple example, as well as other possible examples, shows that: (1) the inflated models and covariates information improve fairness of calculated rate premium; and (2) in the situation that number of reported claims uniformly distributed in past experience of a policyholder (for instance $k_{1}=1$ and $k_{2}=1$ instead of $k_{1}=0$ and $k_{2}=2$). His/Her rate premium under inflated models is more fair and acceptable.

Now, to estimate the pure premium, we consider one mixture Pareto distribution/regression model, as an appropriate model for claim’s severity, along with other well fitted counting models. Moreover, we study situation that total claim size is either 1000 units (Case A) or 5000 unites (Case B). Table 10 and Table 11 show the pure premium under these assumptions.

Table 10: The pure premium for three categories $A_{1},$ $A_{2},$ and $A_{3}$ using well fitted models, whenever total claim size either 1000 or 5000 unites and exact number of reported claim for each year of the policyholder’s experience has been considered.

Case A: Total of reported claim reach to 1000 unites

Model:

Number of cumulated

$NBM_{1}$ & Pareto$M_{1}$

$NBM_{2}$ & Pareto$M_{1}$

$NBM_{3}$ & Pareto$M_{1}$

Year claims up to this year ($K$) $A_{1}$ $A_{2}$ $A_{3}$ $A_{1}$ $A_{2}$ $A_{3}$ $A_{1}$ $A_{2}$ $A_{3}$

$t=0$ — 613.739 723.848 461.546 607.554 730.992 463.354 623.147 726.489 449.722

$K=0$ 589.189 694.894 452.315 574.376 667.716 453.895 594.470 634.635 405.854

$K=1$ 629.269 723.241 460.357 561.898 650.758 441.088 576.836 631.235 423.590

$t=1$ $K=2$ 622.519 707.323 448.918 735.309 652.794 470.045 697.730 620.015 661.912

$K=3$ 621.617 689.808 440.058 2128.105 974.677 900.347 1860.741 811.248 3013.693

$K=4$ 620.905 680.503 432.993 3921.243 2548.529 2535.831 3775.850 2137.017 6735.508

$K=0$ 564.640 665.940 443.084 568.061 641.339 446.372 587.939 605.724 396.865

$K=1$ 601.425 696.455 452.063 533.528 626.073 429.213 553.127 607.359 399.204

$t=2$ $K=2$ 598.391 678.095 441.677 497.293 580.094 413.935 510.093 570.899 416.165

$K=3$ 596.071 663.875 433.633 643.233 573.089 510.020 603.755 555.112 714.869

$K=4$ 598.049 652.537 424.333 1769.746 702.016 1106.136 1514.878 613.404 2298.398

Case B: Total of reported claim reach to 5000 unites

Model:

Number of cumulated

$NBM_{1}$ & Pareto$M_{1}$

$NBM_{2}$ & Pareto$M_{1}$

$NBM_{3}$ & Pareto$M_{1}$

Year claims up to this year ($K$) $A_{1}$ $A_{2}$ $A_{3}$ $A_{1}$ $A_{2}$ $A_{3}$ $A_{1}$ $A_{2}$ $A_{3}$

$t=0$ — 613.739 723.848 461.546 607.554 730.992 463.354 623.147 726.489 449.722

$K=0$ 589.189 694.894 452.315 574.376 667.716 453.895 594.470 634.635 405.854

$K=1$ 799.370 944.429 550.863 713.788 686.966 456.491 732.764 666.356 438.382

$t=1$ $K=2$ 790.796 923.642 537.174 934.074 689.115 486.460 886.338 654.512 685.028

$K=3$ 789.650 900.770 526.572 2703.365 1028.907 931.789 2363.729 856.385 3118.939

$K=4$ 788.745 888.620 518.119 4981.215 2690.327 2624.388 4796.520 2255.920 6970.728

$K=0$ 564.640 665.940 443.084 568.061 641.339 446.372 587.939 605.724 396.865

$K=1$ 763.999 909.450 540.937 677.749 643.612 436.741 702.646 624.374 406.205

$t=2$ $K=2$ 760.145 885.475 528.510 631.719 596.345 421.194 647.979 586.893 423.463

$K=3$ 757.198 866.907 518.885 817.108 589.144 518.964 766.960 570.663 727.407

$K=4$ 759.711 852.102 507.757 2248.135 721.683 1125.535 1924.372 630.588 2338.707

Table 11: The pure premium for three categories $A_{1},$ $A_{2},$ and $A_{3}$ using well fitted models, whenever total claim size either 1000 or 5000 unites and exact number of reported claim for each year of the policyholder’s experience has been considered.

Case A: Total of reported claim reach to 1000 unites

Model:

Year Number of reported

$1INBM_{1}$ & Pareto$M_{1}$

$1INBM_{2}$ & Pareto$M_{1}$

$1INBM_{3}$ & Pareto$M_{1}$

claims at year $l$ ($k_{l}$) $A_{1}$ $A_{2}$ $A_{3}$ $A_{1}$ $A_{2}$ $A_{3}$ $A_{1}$ $A_{2}$ $A_{3}$

$t=0$ – 623.391 757.191 571.461 611.317 733.685 534.525 609.443 790.408 566.132

$k_{1}=0$ 398.970 477.030 502.886 507.393 594.285 513.144 481.452 497.957 543.487

$k_{1}=1$ 1023.795 1085.798 790.796 698.893 882.399 542.755 768.625 943.307 595.197

$t=1$ $k_{1}=2$ 3195.859 2390.932 2178.476 1201.672 1439.796 1069.149 1701.115 1621.325 981.386

$k_{1}=3$ 4019.222 2562.144 2203.502 1405.025 1706.367 1169.205 2099.022 1815.284 915.449

$k_{1}=4$ 4712.021 2973.705 2554.659 1573.213 1920.324 1310.525 2403.626 2078.797 1007.430

$k_{1}=0,k_{2}=0$ 219.378 259.780 332.448 322.694 372.101 366.774 290.423 282.966 388.461

$k_{1}=0,k_{2}=1$ 650.477 721.531 682.960 587.958 712.707 523.542 608.265 614.247 523.977

$k_{1}=0,k_{2}=2$ 2342.976 1565.418 1940.906 1052.665 1190.942 1031.414 1427.979 1142.587 954.742

$k_{1}=1,k_{2}=0$ 650.477 721.531 682.960 587.958 712.707 523.542 608.265 614.247 523.977

$t=2$ $k_{1}=1,k_{2}=1$ 1406.766 1259.673 1035.449 754.650 953.939 545.056 910.456 976.625 559.523

$k_{1}=1,k_{2}=2$ 2936.409 1942.306 2239.077 1170.474 1366.822 1049.038 1661.517 1398.870 937.710

$k_{1}=2,k_{2}=0$ 2342.976 1565.418 1940.906 1052.665 1190.942 1031.414 1427.979 1142.587 954.742

$k_{1}=2,k_{2}=1$ 2936.409 1942.306 2239.077 1170.474 1366.822 1049.038 1661.517 1398.870 937.710

$k_{1}=2,k_{2}=2$ 3520.914 2276.944 2816.357 1392.471 1577.924 1333.877 2008.510 1572.678 1193.225

Case B: Total of reported claim reach to 5000 unites

Model:

Year Number of reported

$1INBM_{1}$ & Pareto$M_{1}$

$1INBM_{2}$ & Pareto$M_{1}$

$1INBM_{3}$ & Pareto$M_{1}$

claims at year $l$ ($k_{l}$) $A_{1}$ $A_{2}$ $A_{3}$ $A_{1}$ $A_{2}$ $A_{3}$ $A_{1}$ $A_{2}$ $A_{3}$

$t=0$ – 623.391 757.191 571.461 611.317 733.685 534.525 609.443 790.408 566.132

$k_{1}=0$ 398.970 477.030 502.886 507.393 594.285 513.144 481.452 497.957 543.487

$t=1$ $k_{1}=1$ 1300.542 1417.866 946.265 887.815 1152.261 649.459 976.396 1231.797 712.211

$k_{1}=2$ 4059.749 3122.146 2606.761 1526.503 1880.126 1279.341 2160.953 2117.171 1174.325

$k_{1}=3$ 5105.682 3345.717 2636.704 1784.825 2228.221 1399.067 2666.421 2370.447 1095.424

$k_{1}=4$ 5985.752 3883.148 3056.902 1998.477 2507.613 1568.174 3053.363 2714.552 1205.490

$k_{1}=0,k_{2}=0$ 219.378 259.780 332.448 322.694 372.101 366.774 290.423 282.966 388.461

$k_{1}=0,k_{2}=1$ 826.311 942.195 817.229 746.892 930.673 626.469 772.688 802.100 626.989

$k_{1}=0,k_{2}=2$ 2976.319 2044.167 2322.484 1337.216 1555.166 1234.188 1813.983 1492.022 1142.443

$k_{1}=1,k_{2}=0$ 826.311 942.195 817.229 746.892 930.673 626.469 772.688 802.100 626.989

$t=2$ $k_{1}=1,k_{2}=1$ 178.7037 1644.916 1239.016 958.644 1245.680 652.213 1156.567 1275.304 669.525

$k_{1}=1,k_{2}=2$ 3730.166 2536.317 2679.275 1486.871 1784.835 1255.277 2110.651 1826.684 1122.062

$k_{1}=2,k_{2}=0$ 2976.319 2044.167 2322.484 1337.216 1555.166 1234.188 1813.983 1492.022 1142.443

$k_{1}=2,k_{2}=1$ 3730.166 253.6317 2679.275 148.6871 1784.835 1255.277 2110.651 1826.684 1122.062

$k_{1}=2,k_{2}=2$ 4472.672 2973.297 3370.048 1768.877 2060.497 1596.115 2551.441 2053.647 1427.811

Same as the above, to illustrate a guideline to use result of Tables 10 and 11, suppose that either Negative Binomial with 2 mixture components, $NBM_{2},$ or 1-Inflated Negative Binomial with 2 mixture components, $1INBM_{2},$ can be considered as an appropriate model for claim frequency. Now consider the following three different scenarios.

Scenario 1:

For a given policyholder in category $A_{1}$ of Table 7. Based upon Table 10’s and Table 11’s result, respectively, his/her second year pure premium under $NBM_{2}$ model is 622.519 units while his/her second year pure premium under $1INBM_{2}$ model is 3195.859 units, whenever such policyholder reported 2 claims with total size 1000 units in the first year. But in the situation that total size of two reported claims reach to 5000 units. He/she has to pay 790.796 units, under $NBM_{2}$ model, and 4059.749 units, under $1INBM_{2}$ model.

Scenario 2:

The given policyholder belongs to category $A_{2}$ of Table 7. Based upon Table 10’s and Table 11’s result, respectively, his second year pure premium, under $NBM_{2}$ model, is 707.323 units while his second year pure premium, under $1INBM_{2}$ model, is 2390.932 units, whenever such policyholder reported 2 claims with total size 1000 units in the first year. But in the situation that total size of two reported claims reach to 5000 units. He has to pay 932.642 units, under $NBM_{2}$ model, and 3122.146 units, under $1INBM_{2}$ model.

Scenario 3:

The given policyholder belongs to category $A_{3}$ of Table 7. Based upon Table 10’s and Table 11’s result, respectively, her second year pure premium, under $NBM_{2}$ model, is 440.918 units while her second year pure premium, under $1INBM_{2}$ model, is 2178.476 units, whenever such policyholder reported 2 claims with total size 1000 units in the first year. But in the situation that total size of two reported claims reach to 5000 units. She has to pay 537.174 units, under $NBM_{2}$ model, and 2606.761 units, under $1INBM_{2}$ model.

The above simple example shows that: (1) the inflated models provides more fair pure premium of policyholders who made some claims in their past experience. While for both cases A and B, the pure premium under non-inflated models do not fairly penalized such policyholders; and (2) in the situation that number of reported claims uniformly distributed in past experience of a policyholder (for instance $k_{1}=1$ and $k_{2}=1$ instead of $k_{1}=0$ and $k_{2}=2$). His/Her pure premium under inflated models is more appealing and acceptable.

7 Conclusion and suggestion

This article introduces an k-Inflated Negative Binomial mixture (kIBNM) distribution/regression model and provides an EM algorithm to estimate its parameters. As an application of the kIBNM distribution/regression to model number of reported claim under a rate–making system has been given. Moreover, in order to compute the pure premium under the system, severity of reported claim has been model with a Pareto mixture distribution/regression model. As an application frequency of reported claim of Iranian third party liability, at 2011, has been model by the kIBNM and all of possible models that have been used by authors. Numerical illustration shows that: (1) the kIBNM models provide more fair rate/pure premiums for policyholders under a rate–making system; and (2) in the situation that number of reported claims uniformly distributed in past experience of a policyholder (for instance $k_{1}=1$ and $k_{2}=1$ instead of $k_{1}=0$ and $k_{2}=2$). The rate/pure premium under the kIBNM models are more appealing and acceptable.

We conjecture that the result of this article may be improved by considering a Double Inflated Negative Binomial with probability mass function $P(Y=y|{\boldsymbol{\theta}})=p_{1}I_{k_{1}}(y)+p_{2}I_{k_{2}}(y)+\sum_{j=3}^{m}p_{j}\left(\genfrac{}{}{0.0pt}{}{y+{\alpha}_{j}-1}{y}\right){\left(\frac{{\tau}_{j}}{{\alpha}_{j}+{\tau}_{j}}\right)}^{{\alpha}_{j}}{\left(\frac{{\alpha}_{j}}{{\alpha}_{j}+{\tau}_{j}}\right)}^{y}I_{{\mathbb{N}}}(y),$ where $k_{1},k_{2}\in{\mathbb{N}},$ $\sum_{j=1}^{m}p_{j}=1,$ and $p_{j},\alpha_{j},\tau_{j}\geq 0,$ for all $j=1,\cdots,m.$