esc.EIGA: Expected severity of claims based on the Exponential-Inverse...

In nlirms: Non-Life Insurance Rate-Making System

Description

esc.EIGA() function gives the expected severity of next claim for a policyholder with regards to the claims severity and freuency history of this policyholder in past time, based on the Exponential-Inverse Gamma (Pareto) model.

Usage

EIGA(x, mu, sigma)
dEIGA(x= 100, claim=1, mu = 50, sigma = 2)
esc.EIGA(sumsev=100, claim =1, mu =50 , sigma = 2)

Arguments

x	vector of quantiles
mu	positive mean parameter of the Exponential-Inverse Gamma (Pareto) distribution that it wil be obtained from fitting Exponential-Inverse Gamma (Pareto) distribution to the claim severity data.
sigma	positive scale parameter of the Exponential-Inverse Gamma (Pareto) distribution that it will be obtained from fitting Exponential-Inverse Gamma (Pareto) distribution to the claim severity data.
sumsev	sum severity of all claims that a policyholder had in past years
claim	total number of claims that a policyholder had in past years

Details

Consider that x be the size of the claim of each insured and z is the mean claim size for each insured, where conditional distribution of the size given the parameter z, is distributed according to exponential(z). if z following the Inverse Gamma distribution, z~IGA(mu, sigma), with Parameterization that E(z)=mu, Then by apply the Bayes theorem the unconditional distribution of claim size x will be exponential-Inverse Gamma (Pareto) model, EIGA~(mu, sigma), with probability density function as the following form:

f(x)=sigma*[(mu*(sigma-1))^sigma]/[(x+mu*(sigma-1))^(sigma+1)]

let claim=k1+ ...+kt, is the total number of claims and sumsev=x1+ ...+xclaim is the total amuont of claim size where xi is the amount of claim size in the claim i, (i=1, ..., i=claim). by apply the Bayes theorem, the posterior structure function of x given the claims size history of the policyholder i.e. f(x|x1, ..., xclaim), following the Inverse Gamma distribution, IGA(claim+sigma, sumsev*mu*(sigma-1). the expected claim severity based on the EIGA model is equal to the mean of this posteriori distribution.

Value

esc.EIGA() function return the expected severity of next claim based on the Exponential-Inverse Gamma (Pareto) model. dEIGA() function return the probability density of Exponential-Inverse Gamma (Pareto) distribution.

Note

esc.EIGA() does not dependent to time. in esc.EIGA() function mu must be grether than 0 and sigma must be grether than 1.

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

esc.EIGA(sumsev=100 ,claim=1 , mu=50, sigma=2)
claim=0:5
esc.EIGA(sumsev=100 ,claim=claim , mu=50, sigma=2)

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