betaII                 package:VGAM                 R Documentation

_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 3-parameter  beta II
     distribution.

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

     betaII(link.scale = "loge", link.p = "loge", link.q = "loge",
            init.scale = NULL, init.p = 1, init.q = 1, zero = NULL)

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

link.scale, link.p, link.q: Parameter link functions applied to the
          (positive) parameters 'scale', 'p' and 'q'. See 'Links' for
          more choices.

init.scale, init.p, init.q: Optional initial values for 'scale', '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} which correspond to 'scale',
          'p', 'q', respectively.

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

     The 3-parameter beta II is the 4-parameter _generalized_ beta II
     distribution with shape parameter a=1. It is also known as the
     Pearson VI distribution. Other distributions which are special
     cases of the 3-parameter beta II include the Lomax (p=1) and
     inverse Lomax (q=1). More details can be found in Kleiber and Kotz
     (2003).

     The beta II distribution has density

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

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

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

     provided q > 1.

_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'.

_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:

     'betaff', 'genbetaII', 'dagum', 'sinmad', 'fisk', 'invlomax',
     'lomax', 'paralogistic', 'invparalogistic'.

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

     y = rsinmad(n=2000, a=1, 6, 2)  # Not genuine data!
     fit = vglm(y ~ 1, betaII, trace=TRUE)
     fit = vglm(y ~ 1, betaII(init.p=0.7, init.q=0.7), trace=TRUE, crit="c")
     coef(fit, mat=TRUE)
     Coef(fit)
     summary(fit)

