enc.PGA: Expected number of claims based on the Poisson-Gamma...

In nlirms: Non-Life Insurance Rate-Making System

Description

enc.PGA() 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-Gamma (Negetive Binomial) model.

Usage

PGA(k, mu, sigma)
dPGA(k=1, mu=.1, sigma=2)
enc.PGA(time = 2, claim = 1, mu = .1, sigma = 2)

Arguments

k	vector of (non-negative integer) quantiles.
mu	positive mean parameter of the Poisson-Gamma (Negetive Binomial) distribution that it wil be obtained from fitting Poisson-Gamma (Negetive Binomial) distribution to the claim frequency data.
sigma	positive scale parameter of the Poisson-Gamma (Negetive Binomial) distribution that it will be obtained from fitting Poisson-Gamma (Negetive Binomial) 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 Gamma distribution, y~GA(mu, sigma), with Parameterization that E(y)=mu, Then by apply the Bayes theorem the unconditional distribution of the number of claims k will be Poisson-Gamma (Negetive Binomial) distribution, PIGA~(mu, sigma), with probability density function as the following form:

f(y)=[gamma(sigma+k)/(gamma(k+1)*gamma(sigma))]*[mu/(mu+sigma)]^k*[sigma/(mu+sigma)]^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 Gamma distribution, GA(time+(sigma/mu), claim+sigma). the expected number of claims based on the PGA model is equal to the mean of this posteriori distribution.

Value

enc.PGA() function return the expected number of claims based on the Poisson-Gamma (Negetive Binomial) model. dPGA() function return the probability density of Poisson-Gamma (Negative Binomial) model.

Note

in enc.PGA() function mu and sigma must be grether than 0.

Author(s)

Saeed Mohammadpour ([e-mail]), Soodabeh Mohammadpoor Golojeh ([e-mail])

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.

Examples

dPGA(k=1, mu=.1, sigma=2)
enc.PGA(time = 2, claim = 1, mu = .1, sigma = 2)
time=1:5
enc.PGA(time = time, claim = 1, mu = .1, sigma = 2)

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