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

Description Usage Arguments Details Value Note Author(s) References Examples

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.

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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)

`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.

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.

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.

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.

Saeed Mohammadpour (s.mohammadpour1111@gmail.com), Soodabeh Mohammadpoor Golojeh (s.mohammadpour@gmail.com)

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.

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)