esc.EGIG: Expected severity of claims based on the...
In nlirms: Non-Life Insurance Rate-Making System

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

esc.EGIG() 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-Generalized Inverse Gaussian model.

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EGIG(x, mu, sigma, nu)
dEGIG(x= 100, claim=1, mu = 50, sigma = 2, nu=2)
esc.EGIG(sumsev=100 ,claim=1 , mu=50, sigma=2, nu=2)

`x`	vector of quantiles
`mu`	positive mean parameter of the Exponential-Generalized Inverse Gaussian distribution that it wil be obtained from fitting Exponential-Generalized Inverse Gaussian distribution to the claim severity data.
`sigma`	positive dispersion parameter of the Exponential-Generalized Inverse Gaussian distribution that it will be obtained from fitting Exponential-Generalized Inverse Gaussian distribution to the claim severity data.
`nu`	third parameter of the Exponential-Generalized Inverse Gaussian distribution that it will be obtained from fitting Exponential-Generalized Inverse Gaussian 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

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 Generalized Inverse Gaussian distribution, z~GIG(mu, sigma, nu), with Parameterization that E(z)=mu, Then by apply the Bayes theorem the unconditional distribution of claim size x will be Exponential-Generalized Inverse Gaussian model, EGIG~(mu, sigma, nu), with probability density function as the following form:

f(y)=[c*(sigma*alpha)^((nu+1)/2) * besselK(alpha,nu-1)] / [mu*besselK(1/sigma,nu)]

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

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

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 Generalized Inverse Guassian distribution, GIG(c/(mu*sigma), [mu/(c*sigma)]+2*sumsev, nu-claim). the expected claim severity based on the EGIG model is equal to the mean of this posteriori distribution.

esc.EGIG() function return the expected severity of next claim based on the Exponential-Generalized Inverse Gaussian model. dEIGA() function return the probability density of Exponential-Generalized Inverse Gaussian distribution.

esc.EGIG() does not dependent to time. in esc.EGIG() function mu and sugma must be grether than 0 and -inf<nu<inf.

esc.EGIG() function for nu=-.5, return the expected severity of next claim based on the Exponential-Inverse Gaussian model and dEIGA() function return the probability density of Exponential-Inverse Gaussian distribution.

esc.EGIG() function for nu=0, return the expected severity of next claim based on the Exponential-Harmonic model and dEIGA() function return the probability density of Exponential-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.

claim=0:5
# Expected severity of claims based on the Exponential-Generalized Inverse Gaussian model
esc.EGIG(sumsev=100 ,claim=1 , mu=50, sigma=2, nu=1)

# Expected severity of claims based on the Exponential-Inverse Gaussian model
esc.EGIG(sumsev=100 ,claim=1 , mu=50, sigma=2, nu=-.5)

# Expected severity of claims based on the Exponential-Harmonic model
esc.EGIG(sumsev=100 ,claim=claim , mu=50, sigma=2, nu=0)