sinmad                 package:VGAM                 R Documentation

_S_i_n_g_h-_M_a_d_d_a_l_a _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:

     Maximum likelihood estimation of the 3-parameter  Singh-Maddala
     distribution.

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

     sinmad(link.a = "loge", link.scale = "loge", link.q = "loge",
            init.a = NULL, init.scale = NULL, init.q = 1, zero = NULL)

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

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

init.a, init.scale, init.q: Optional initial values for 'a', 'scale',
          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 'a',
          'scale', 'q', respectively.

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

     The 3-parameter Singh-Maddala distribution is the 4-parameter
     generalized beta II distribution with shape parameter p=1. It is
     known under various other names, such as the Burr XII (or just the
     Burr distribution), Pareto IV, beta-P, and generalized
     log-logistic distribution. More details can be found in Kleiber
     and Kotz (2003).

     Some distributions which are special cases of the 3-parameter
     Singh-Maddala are the Lomax (a=1), Fisk (q=1), and paralogistic
     (a=q).

     The Singh-Maddala distribution has density

            f(y) = aq y^(a-1) / [b^a (1 + (y/b)^a)^(1+q)]

     for a > 0, b > 0, q > 0, y > 0. Here, b is the scale parameter
     'scale', and the others are shape parameters. The cumulative
     distribution function is

                    F(y) = 1 - [1 + (y/b)^a]^(-q).

     The mean is

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

     provided -a < 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'.

_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:

     'Sinmad', 'genbetaII', 'betaII', 'dagum', 'fisk', 'invlomax',
     'lomax', 'paralogistic', 'invparalogistic'.

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

     y = rsinmad(n=3000, 3, 5, 2)
     fit = vglm(y ~ 1, sinmad, trace=TRUE)
     fit = vglm(y ~ 1, sinmad, trace=TRUE, crit="c")
     coef(fit, mat=TRUE)
     Coef(fit)
     summary(fit)

