enc.PIGA() 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-Inverse Gamma model.
PIGA(k, mu, sigma)
dPIGA(k = 1, mu = .1, sigma = 2)
enc.PIGA(time = 2, claim = 1, mu = .1, sigma = 2)vector of (non-negative integer) quantiles.
positive mean parameter of the Poisson-Inverse Gamma distribution that it wil be obtained from fitting Poisson-Inverse Gamma distribution to the claim frequency data.
positive scale parameter of the Poisson-Inverse Gamma distribution that it will be obtained from fitting Poisson-Inverse Gamma 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.PIGA() function return the expected number of claims based on the Poisson-Inverse Gamma model. dPIGA() function return the probability density of Poisson-Inverse Gamma distribution.
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 Inverse Gamma distribution, y~IGA(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-Inverse Gamma distribution, PIGA~(mu, sigma), with probability density function as the following form:
f(y)=2*alpha^[(k+sigma)/2]*besselK((4*alpha)^.5,k-sigma)/ [gamma(k+1)*gamma(sigma]
where alpha=mu*(sigma-1).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 Generalized Inverse Gaussian distribution, GIG(2*time, 2*mu*(sigma-1), claim-sigma-1). the expected number of claims based on the PIGA model is equal to the mean of this posteriori distribution.
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# NOT RUN {
dPIGA(k=1, mu=.1, sigma=2)
enc.PIGA(time = 2, claim = 1, mu = .1, sigma = 2)
time=1:5
enc.PIGA(time = time, claim = 1, mu = .1, sigma = 2)
# }
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