# enc.PGIG: Expected number of claims based on the Poisson-Generalized... In nlirms: Non-Life Insurance Rate-Making System

## Description

enc.PGIG() function gives the expected number of claims for a policyholder in the next time (for example in next year) with regards to the number of claims history of this policyholder in past time, based on the Poisson-Generalized Inverse Gaussian (Sichel) model.

## Usage

 ```1 2 3``` ```PGIG(k, mu, sigma, nu) dPGIG(k = 1, mu = .1, sigma = 2, nu=1) enc.PGIG(time = 2, claim = 1, mu =.1, sigma = 2,nu=1) ```

## Arguments

 `k` vector of (non-negative integer) quantiles. `mu` positive mean parameter of the Poisson-Generalized Inverse Gaussian (Sichel) distribution that it wil be obtained from fitting Poisson-Generalized Inverse Gaussian (Sichel) distribution to the claim frequency data. `sigma` positive dispersion parameter of the Poisson-Generalized Inverse Gaussian (Sichel) distribution that it will be obtained from fitting Poisson-Generalized Inverse Gaussian (Sichel) distribution to the claim frequency data. `nu` third parameter of the Poisson-Generalized Inverse Gaussian (Sichel) distribution that it will be obtained from fitting Poisson-Generalized Inverse Gaussian (Sichel) distribution to the claim frequency data. `time` time period to claims freuency rate-making. `claim` total number of claims that a policyholder had in past years.

## Details

Consider that the number of claims k, (k=0,1,...), given the parameter y, is distributed according to Poisson(y), where y is denoting the different underlyin risk of each policyholder to have an accident. if y following the Generalized Inverse Gaussian distribution, y~GIG(mu, sigma, nu), with Parameterization that E(y)=mu, Then by apply the Bayes theorem the unconditional distribution of the number of claims k will be Poisson-Generalized Inverse Gaussian (Sichel) distribution, PGIG~(mu, sigma, nu), with probability density function as the following form:

f(y)=[(mu/c)^nu * besselK(alpha,k+nu)] / [gamma(k+1)*(alpha*sigma)^(k+nu)*besselK(1/sigma,nu)]

where c=besselK(1/sigma,nu+1)/besselK(1/sigma,nu)

alpha^2=[1/(sigma^2)]+[2*mu/(c*sigma)].

let claim=k1+ ...+kt, is total number of claims that a policyholder had in t years, where ki is the number of claims that the policyholder had in the year i, (i=1, ..., t=time). by apply the Bayes theorem, the posterior structure function of y i.e. f(y|k1, ..., kt), for a policyholder with claim history k1,..., kt, following the Generalized Gaussian distribution, GIG(2*time+[c/(sigma*mu)], c/(sigma*mu), nu+claim). the expected number of claims based on the PGA model is equal to the mean of this posteriori distribution.

## Value

enc.PGIG() function return the expected number of claims based on the Poisson-Generalized Inverse Gaussian (Sichel) model. dPGIG() function return the probability density of Poisson-Generalized Inverse Gaussian (Sichel) distribution.

## Note

in enc.PGIG() function mu and sigma must be grether than 0 and -inf<nu<inf.

enc.PGIG() function for nu=-.5, return the expected number of claims based on the Poisson-Inverse Gaussian model and dPGIG() function return the probability density of Poisson-Inverse Gaussian distribution.

enc.PGIG() function for nu=0, return the expected number of claims based on the Poisson-Harmonic model and dPGIG() function return the probability density of Poisson-Harmonic distribution.

## References

Frangos, N. E., & Vrontos, S. D. (2001). Design of optimal bonus-malus systems with a frequency and a severity component on an individual basis in automobile insurance. ASTIN Bulletin: The Journal of the IAA, 31(1), 1-22.

Lemaire, J. (1995) Bonus-Malus Systems in Automobile Insurance, Kluwer Academic Publishers, Massachusetts.

MohammadPour, S., Saeedi, K., & Mahmoudvand, R. (2017). Bonus-Malus System Using Finite Mixture Models. Statistics, Optimization & Information Computing, 5(3), 179-187.

Najafabadi, A. T. P., & MohammadPour, S. (2017). A k-Inflated Negative Binomial Mixture Regression Model: Application to Rate–Making Systems. Asia-Pacific Journal of Risk and Insurance, 12.

Rigby, R. A., & Stasinopoulos, D. M. (2005). Generalized additive models for location, scale and shape. Journal of the Royal Statistical Society: Series C (Applied Statistics), 54(3), 507-554.

Stasinopoulos, D. M., Rigby, B. A., Akantziliotou, C., Heller, G., Ospina, R., & Motpan, N. (2010). gamlss. dist: Distributions to Be Used for GAMLSS Modelling. R package version, 4-0.

Stasinopoulos, D. M., & Rigby, R. A. (2007). Generalized additive models for location scale and shape (GAMLSS) in R. Journal of Statistical Software, 23(7), 1-46.

Tzougas, G., & Fragos, N. (2013). Design of an optimal Bonus-Malus system using the Sichel distribution as a model of claim counts.

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10``` ```time=1:5 claim=0:4 # Expected number of claims based on the Poisson-Generalized Inverse Gaussian model enc.PGIG(time = time, claim = claim, mu = .1, sigma = 2, nu=1) # Expected number of claims based on the Poisson-Inverse Gaussian model enc.PGIG(time = time, claim = claim, mu = .1, sigma = 2, nu=-.5) # Expected number of claims based on the Poisson-Harmonic model enc.PGIG(time = time, claim = claim, mu = .1, sigma = 2, nu=0) ```

nlirms documentation built on May 1, 2019, 7:06 p.m.