panjer                package:actuar                R Documentation

_P_a_n_j_e_r'_s _r_e_c_u_r_i_s_o_n _f_o_r_m_u_l_a

_D_e_s_c_r_i_p_t_i_o_n:

     Panjer recursion formula to compute the total amount of claims
     probability function of a portfolio.

_U_s_a_g_e:

     panjer(fx, freq.dist = c("poisson", "negative binomial", "binomial",
     "geometric", "logarithmic"),par, p0, TOL = 1e-08, echo= FALSE)

_A_r_g_u_m_e_n_t_s:

      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. 

_D_e_t_a_i_l_s:

     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.

_V_a_l_u_e:

     A vector representing the probability density function of the
     total amount of claims.

_A_u_t_h_o_r(_s):

     Vincent Goulet vincent.goulet@act.ulaval.ca and Sbastien Auclair

_R_e_f_e_r_e_n_c_e_s:

     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

_E_x_a_m_p_l_e_s:

     ### (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)

