mccullagh89               package:VGAM               R Documentation

_M_c_C_u_l_l_a_g_h (_1_9_8_9) _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:

     Estimates the two parameters of the McCullagh (1989) distribution
     by maximum likelihood estimation.

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

     mccullagh89(ltheta="rhobit", lnu="logoff",
                 itheta=NULL, inu=NULL,
                 etheta=list(),
                 enu=if(lnu == "logoff") list(offset=0.5) else list(),
                 zero=NULL)

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

ltheta, lnu: Link functions for the theta and nu parameters. See
          'Links' for more choices.

itheta, inu: Numeric. Optional initial values for theta and nu. The
          default is to internally compute them.

etheta, enu: List. Extra argument associated with 'ltheta' and 'lnu'
          containing any extra information. See 'Links' for general
          information about 'VGAM' link functions.

    zero: An integer-valued vector specifying which linear/additive
          predictors are modelled as intercepts only. The default is
          none of them. If used, choose one value from the set {1,2}.

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

     The McCullagh (1989) distribution has density function

 f(y;theta,nu) =  (1-y^2)^(nu-0.5) / [ (1 - 2*theta*y+theta^2)^nu * Beta(nu+0.5, 0.5)]

     where -1 < y < 1 and -1 < theta < 1. This distribution is equation
     (1) in that paper. The parameter nu satisfies nu > -1/2, therefore
     the default is to use an log-offset link with offset equal to 0.5,
     i.e., eta_2=log(nu+0.5). The mean is of Y is nu*theta/(1+nu), and
     these are returned as the fitted values.

     This distribution is related to the Leipnik distribution (see
     Johnson et al. (1995)), is related to ultraspherical functions,
     and under certain conditions, arises as exit distributions for
     Brownian motion. Fisher scoring is implemented here and it uses a
     diagonal matrix so the parameters are globally orthogonal in the
     Fisher information sense. McCullagh (1989) also states that, to
     some extent, theta and nu have the properties of a location
     parameter and a precision parameter, respectively.

_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'.

_N_o_t_e:

     Convergence may be slow or fail unless the initial values are
     reasonably close. If a failure occurs, try assigning the argument
     'inu' and/or 'itheta'.  Figure 1 of McCullagh (1989) gives a broad
     range of densities for different values of theta and nu, and this
     could be consulted for obtaining reasonable initial values if all
     else fails.

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

     T. W. Yee

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

     McCullagh, P. (1989) Some statistical properties of a family of
     continuous univariate distributions. _Journal of the American
     Statistical Association_, *84*, 125-129.

     Johnson, N. L. and Kotz, S. and Balakrishnan, N. (1995)
     _Continuous Univariate Distributions_, 2nd edition, Volume 2, New
     York: Wiley. (pages 612-617).

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

     'leipnik', 'rhobit', 'logoff'.

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

     n = 1000
     y = rnorm(n, mean=0.0, sd=0.2)  # Limit as theta is 0, nu is infinity
     fit = vglm(y ~ 1, mccullagh89, trace=TRUE)
     fitted(fit)[1:5]
     mean(y)
     summary(fit)
     coef(fit, matrix=TRUE)
     Coef(fit)

