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.
PGA(k, mu, sigma)
dPGA(k=1, mu=.1, sigma=2)
enc.PGA(time = 2, claim = 1, mu = .1, sigma = 2)vector of (non-negative integer) quantiles.
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.
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 period to claims freuency rate-making
total number of claims that a policyholder had in past years
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.
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.
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# NOT RUN {
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)
# }
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