benini                 package:VGAM                 R Documentation

_B_e_n_i_n_i _D_i_s_t_r_i_b_u_t_i_o_n _F_a_m_i_l_y _F_u_n_c_t_i_o_n

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

     Estimating the parameter of the Benini distribution by maximum
     likelihood estimation.

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

     benini(y0=stop("argument \"y0\" must be specified"),
            lshape="loge", ishape=NULL, method.init=1)

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

      y0: Positive scale parameter.  

  lshape: Parameter link function applied to the parameter b, which is
          the shape parameter. See 'Links' for more choices. A log link
          is the default because b is positive.

  ishape: Optional initial value for the shape parameter. The default
          is to compute the value internally.

method.init: An integer with value '1' or '2' which specifies the
          initialization method. If failure to converge occurs try the
          other value, or else specify a value for 'ishape'.

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

     The Benini distribution has a probability density function that
     can be written

         f(y) = 2*b*exp(-b * [(log(y/y0))^2]) * log(y/y0) / y

     for y_0>0, y0<y, and b>0. The cumulative distribution function for
     Y is

                F(y) = 1 - exp(-b * [(log(y/y0))^2]).

     Here, Newton-Raphson and Fisher scoring coincide.

     On fitting, the 'extra'  slot has a component called 'y0' which 
     contains the value of the 'y0' argument.

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

     An object of class '"vglmff"' (see 'vglmff-class'). The object is
     used by modelling functions such as 'vglm', 'rrvglm' and 'vgam'.

_W_a_r_n_i_n_g:

     The mean of Y, which are returned as the fitted values, may be
     incorrect.

_N_o_t_e:

     Yet to do: the 2-parameter Benini distribution estimates y0 as
     well, and the 3-parameter Benini distribution estimates another
     shape parameter a too.

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

     T. W. Yee

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

     Kleiber, C. and Kotz, S. (2003) _Statistical Size Distributions in
     Economics and Actuarial Sciences_, Hoboken, NJ:
     Wiley-Interscience.

_S_e_e _A_l_s_o:

     'Benini'.

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

     y  = rbenini(n <- 3000, y0=1, shape=exp(2))
     fit = vglm(y ~ 1, benini(y0=1), trace=TRUE, crit="c")
     coef(fit, matrix=TRUE)
     Coef(fit)
     fit@extra$y0

     # Apparent discrepancy:
     fitted(fit)[1:5]
     mean(y)

