panjer {actuar} | R Documentation |
Panjer recursion formula to compute the total amount of claims probability function of a portfolio.
panjer(fx, freq.dist = c("poisson", "negative binomial", "binomial", "geometric", "logarithmic"),par, p0, TOL = 1e-08, echo= FALSE)
fx |
a vector of the (discretized) claim amount distribution; first element *must* be f_X(0) |
freq.dist |
name of the counting distribution |
par |
named list of the parameter(s) of the counting distribution as they are defined in "rdist". |
p0 |
arbitrary amount of probability at zero given to the frequency distribution. It creates zero-modified or zero-truncated distributions |
TOL |
stop recursion when cumulative probability function is less than TOL away from 1 |
echo |
print the cumulative distribution of the total amount of claims as it is computed. |
The formula of the (a, b, 1) class is:
([p1-(a+b)*p0]*fx[x+1] + sum(y=1 to min(x,m);(a+b*y/x)fx[y+1]*fs(x-y)))/(1-a*fx[1])
For the (a, b, 0) class, the result reduces to
sum(y=1 to min(x,m);(a+b*y/x)fx[y+1]*fs(x-y))/(1-a*fx[1])
The counting variable is a member of the (a, b, 0) family of discrete distributions if p0 is not specified and a member of the (a,b,1) family if p0 is specified. The logarithmic distribution is a limiting case of the negative binomial distribution where the size parameter is equal to 0.
A vector representing the probability density function of the total amount of claims.
Vincent Goulet vincent.goulet@act.ulaval.ca and Sébastien Auclair
Klugman, S.A and Panjer,H.H and Willmot, G.E (2004), Loss Models: from data to decision, Second Edition, Wiley, Sections 4.6.5 - 4.6.7, 6.6 and Appendix B
### (a,b,0) class with the binomial distribution (a <- panjer(fx=rep(0.5,2), freq.dist="bin",par=list(size=3, prob=0.5),echo=TRUE)) sum(a) ### Example 6.18 of "Loss Models" (Second Edition). ### (a,b,1) class with the Extended Truncated Negative Binomial distribution. (a <- panjer(fx=c(0.3, 0.5, 0.2), "negative bin", par=list(size=0.2, prob=0.25), p0=0, TOL=1E-5, echo=TRUE)) plot(a)