frank                  package:VGAM                  R Documentation

_F_r_a_n_k'_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 Frank's bivariate
     distribution using maximum likelihood estimation.

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

     frank(lapar="loge", eapar=list(), iapar=2)

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

   lapar: Link function applied to the (positive) association parameter
          alpha. See 'Links' for more choices.

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

   iapar: Numeric. Initial value for alpha. If a convergence failure
          occurs try assigning a different value.

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

     The cumulative distribution function is

 P(Y1 <= y1, Y2 <= y2) =  H_{alpha}(y1,y2) =  log_{alpha} [1 + (alpha^(y1)-1)*(alpha^(y2)-1)/ (alpha-1)]

     for alpha != 1. Note the logarithm here is to base alpha. The
     support of the function is the unit square.

     When 0<alpha<1 the probability density function 
     h_{alpha}(y_1,y_2) is symmetric with respect to the lines y2=y1
     and y2=1-y1. When alpha>1 then h_{1/alpha}(1-y_1,y_2).

     If alpha=1 then H(y1,y2)=y1*y2, i.e., uniform on the unit square.
     As alpha approaches 0 then H(y1,y2)=min(y1,y2). As alpha
     approaches infinity then H(y1,y2)=max(0,y1+y2-1).

     A variant of Newton-Raphson is used, which only seems to work for
     an intercept model.

_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 matrix with two columns and values equal to a half.
     This is 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:

     Genest, C. (1987) Frank's family of bivariate distributions.
     _Biometrika_, *74*, 549-555.

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

     'rfrank', 'fgm'.

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

     ymat = rfrank(n=2000, alpha=exp(4))
     ## Not run: plot(ymat)
     fit = vglm(ymat ~ 1, fam=frank, trace=TRUE)
     coef(fit, matrix=TRUE)
     Coef(fit)
     vcov(fit)
     fitted(fit)[1:5,]
     summary(fit)

