genbetaII                package:VGAM                R Documentation

_G_e_n_e_r_a_l_i_z_e_d _B_e_t_a _D_i_s_t_r_i_b_u_t_i_o_n _o_f _t_h_e _S_e_c_o_n_d _K_i_n_d

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

     Maximum likelihood estimation of the 4-parameter  generalized beta
     II distribution.

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

     genbetaII(link.a = "loge", link.scale = "loge",
               link.p = "loge", link.q = "loge",
               earg.a=list(), earg.scale=list(), earg.p=list(), earg.q=list(),
               init.a = NULL, init.scale = NULL, init.p = 1, init.q = 1,
               zero = NULL)

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

link.a, link.scale, link.p, link.q: Parameter link functions applied to
          the shape parameter 'a', scale parameter 'scale', shape
          parameter 'p', and shape parameter 'q'. All four parameters
          are positive. See 'Links' for more choices.

earg.a, earg.scale, earg.p, earg.q: List. Extra argument for each of
          the links. See 'earg' in 'Links' for general information.

init.a, init.scale: Optional initial values for 'a' and 'scale'. A
          'NULL' means a value is computed internally.

init.p, init.q: Optional initial values for 'p' and 'q'.

    zero: An integer-valued vector specifying which linear/additive
          predictors are modelled as intercepts only. Here, the values
          must be from the set {1,2,3,4} which correspond to 'a',
          'scale', 'p', 'q', respectively.

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

     This distribution is most useful for unifying a substantial number
     of size distributions. For example, the Singh-Maddala, Dagum, Fisk
     (log-logistic), Lomax (Pareto type II), inverse Lomax, beta
     distribution of the second kind distributions are all special
     cases. Full details can be found in Kleiber and Kotz (2003), and
     Brazauskas (2002).

     The 4-parameter generalized beta II distribution has density

       f(y) = a y^(ap-1) / [b^(ap) B(p,q) (1 + (y/b)^a)^(p+q)]

     for a > 0, b > 0, p > 0, q > 0, y > 0. Here B is the beta
     function, and  b is the scale parameter 'scale', while the others
     are shape parameters. The mean is 

   E(Y) = b  gamma(p + 1/a)  gamma(q - 1/a) / ( gamma(p)  gamma(q))

     provided -ap < 1 < aq.

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

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

_N_o_t_e:

     If the self-starting initial values fail, try experimenting with
     the initial value arguments, especially those whose default value
     is not 'NULL'.

     Successful convergence depends on having very good initial values.
     This is rather difficult for this distribution! More improvements
     could be made here.

_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.

     Brazauskas, V. (2002) Fisher information matrix for the
     Feller-Pareto distribution. _Statistics & Probability Letters_,
     *59*, 159-167.

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

     'lino', 'betaff', 'betaII', 'dagum', 'sinmad', 'fisk', 'lomax',
     'invlomax', 'paralogistic', 'invparalogistic'.

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

     y = rsinmad(n=3000, 4, 6, 2) # Not very good data!
     fit = vglm(y ~ 1, genbetaII, trace=TRUE)
     fit = vglm(y ~ 1, genbetaII(init.p=1.0, init.a=4, init.sc=7, init.q=2.3),
                trace=TRUE, crit="c")
     coef(fit, mat=TRUE)
     Coef(fit)
     summary(fit)

