bstraub                package:actuar                R Documentation

_B_u_h_l_m_a_n_n-_S_t_r_a_u_b _C_r_e_d_i_b_i_l_i_t_y _M_o_d_e_l

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

     'bstraub' calculates credibility premiums in the Bhlmann-Straub
     credibility model.

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

     bstraub(ratios, weights,
             heterogeneity = c("iterative", "unbiased"),
             TOL = 1e-06, echo = FALSE)

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

  ratios: matrix of ratios (contracts in lines, years in columns)

 weights: matrix of weights corresponding to ratios

heterogeneity: estimator of the between contract heterogeneity
          parameter used in premium calculation; '"iterative"' for the
          Bischel-Straub estimator; '"unbiased"' for the usual
          Bhlmann-Straub estimator (see below)

     TOL: maximum relative error in the iterative procedure

    echo: boolean, whether to echo iterative procedure or not

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

     The credibility premium of contract i is given by

                    z[i] X[iw] + (1 - z[i]) X[zw],

     where

                   z[i] = (w[i.] a)/(w[i] a + s^2),

     X[iw] is the weighted average of the ratios of contract i, X[zw]
     is the weighted average of the matrix of ratios using credibility
     factors and w[i.] is the total weight of a contract. s^2 is the
     estimator of the within contract heterogeneity and a is the
     estimator of the between contract heterogeneity.

     Missing data are represent by 'NA' in both the matrix of ratios
     and the matrix of weights. The function can cope with complete
     lines of 'NA' in case a contract has no experience.

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

     A list with the following components: 

premiums: vector of credibility premiums

individual: vector of contract weighted averages

collective: collective premium estimator

 weights: vector of contracts total weights, as used in credibility
          factors

      s2: estimator of the within contract heterogeneity parameter

unbiased: unbiased estimator of the between contract heterogeneity
          parameter

iterative: iterative estimator of the between contract heterogeneity
          parameter

_E_s_t_i_m_a_t_i_o_n _o_f _a:

     The Bhlmann-Straub unbiaised estimator ('heterogeneity =
     "unbiased"') of the between contracts heterogeneity parameter is

        a = c sum(w[i.] * (X[iw] - X[ww])^2 - (I - 1) * s^2),

     where c = w[..]/(w[..]^2 - sum(w[i.]^2)) and I is the number of
     contracts.

     The Bishel-Straub pseudo-estimator ('heterogeneity = "iterative"')
     is obtained recursively as the solution of

             a = 1/(I - 1) sum(z[i] * (X[iw] - X[zw])^2).

     The fixed point algorithm is used up, with a relative error of
     'TOL' stopping criteria.

_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:

     Goulet, V. (1998), _Principles and Application of Credibility
     Theory_, Journal of Actuarial Practice, Volume 6, ISSN 1064-6647.

     Goovaerts, M. J. and Kaas, R. and van Heerwaarden, A. E. and
     Bauwelinckx, T. (1990), _Effective actuarial methods_,
     North-Holland.

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

     data(hachemeister)

     ## Credibility premiums calculated with the iterative estimator
     bstraub(hachemeister$claims, hachemeister$weights)

     ## Credibility premiums calculated with the unbiased estimator
     bstraub(hachemeister$claims, hachemeister$weights, heterogeneity = "unbiased")

