plackett                package:VGAM                R Documentation

_P_l_a_c_k_e_t_t'_s _B_i_v_a_r_i_a_t_e _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:

     Estimate the association parameter of Plackett's bivariate
     distribution by maximum likelihood estimation.

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

     plackett(link="loge", earg=list(), ioratio=NULL, method.init=1, nsimEIM=200)

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

    link: Link function applied to the (positive) odds ratio psi. See
          'Links' for more choices.

    earg: List. Extra argument for the link. See 'earg' in 'Links' for
          general information.

 ioratio: Numeric. Optional initial value for psi. If a convergence
          failure occurs try assigning a value or a different value.

method.init: An integer with value '1' or '2' which specifies the
          initialization method for the parameter. If failure to
          converge occurs try another value and/or else specify a value
          for 'ioratio'.

 nsimEIM: See 'CommonVGAMffArguments'.

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

     The defining equation is

                psi = H*(1-y1-y2+H) / ((y1-H)*(y2-H))

     where P(Y1 <= y1, Y2 <= y2)= H(y1,y2) is the cumulative
     distribution function. The density function is h(y1,y2) =

 psi*[1 + (psi-1)*(y1 + y2 - 2*y1*y2) ] / ( [1 + (psi-1)*(y1 + y2)]^2 - 4*psi*(psi-1)*y1*y2)^(3/2)

     for psi > 0. Some writers call psi the _cross product ratio_ but
     it is called the _odds ratio_ here. The support of the function is
     the unit square. The marginal distributions here are the standard
     uniform although it is commonly generalized to other
     distributions.

     If psi=1 then h(y1,y2) = y1*y2, i.e., independence. As the odds
     ratio tends to infinity one has y1=y2. As the odds ratio tends to
     0 one has y2=1-y1.

     Fisher scoring is implemented using 'rplack'. Convergence is often
     quite slow.

_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:

     The response must be a two-column matrix.  Currently, the fitted
     value is a 2-column matrix with 0.5 values because the marginal
     distributions correspond to a standard uniform distribution.

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

     T. W. Yee

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

     Plackett, R. L. (1965) A class of bivariate distributions.
     _Journal of the American Statistical Association_, *60*, 516-522.

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

     'rplack', 'frank'.

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

     ymat = rplack(n=2000, oratio=exp(2))
     ## Not run: plot(ymat, col="blue")
     fit = vglm(ymat ~ 1, fam=plackett, trace=TRUE)
     coef(fit, matrix=TRUE)
     Coef(fit)
     vcov(fit)
     head(fitted(fit))
     summary(fit)

