esc.EGA() 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-Gamma model.
EGA(x, mu, sigma)
dEGA(x= 100, claim=1, mu = 50, sigma = 2)
esc.EGA(sumsev=100, claim =1, mu =50 , sigma = 2)vector of quantiles
positive mean parameter of the Exponential-Gamma distribution that it wil be obtained from fitting Exponential-Gamma distribution to the claim severity data.
positive scale parameter of the Exponential-Gamma distribution that it will be obtained from fitting Exponential-Gamma distribution to the claim severity data.
sum severity of all claims that a policyholder had in past years
total number of claims that a policyholder had in past years
esc.EGA() function return the expected severity of next claim based on the Exponential-Gamma model. dEGA() function return the probability density of Exponential-Gamma distribution.
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 Gamma distribution, z~GA(mu, sigma), with Parameterization that E(z)=mu, Then by apply the Bayes theorem the unconditional distribution of claim size x will be exponential-Gamma model, EGA~(mu, sigma), with probability density function as the following form:
f(x)=2*[(sigma*x/mu)^((sigma+1)/2)]*besselK((4*x*sigma/mu)^.5,sigma-1)/ gamma(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(2*sigma/mu, 2*sumsev, sigma-claim). the expected claim severity based on the EGIG model is equal to the mean of this posteriori distribution.
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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.
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# NOT RUN {
esc.EGA(sumsev=100 ,claim=1 , mu=50, sigma=2)
claim=0:5
esc.EGA(sumsev=100 ,claim=claim , mu=50, sigma=2)
# }
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