bilogistic4               package:VGAM               R Documentation

_B_i_v_a_r_i_a_t_e _L_o_g_i_s_t_i_c _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 four parameters of the bivariate logistic
     distribution by maximum likelihood estimation.

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

     bilogistic4(llocation="identity", lscale="loge",
                 iloc1=NULL, iscale1=NULL, iloc2=NULL, iscale2=NULL,
                 method.init=1, zero=NULL)

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

llocation: Link function applied to both location parameters l1 and l2.
          See 'Links' for more choices.

  lscale: Parameter link function applied to both (positive) scale
          parameters s1 and s2. See 'Links' for more choices. 

iloc1, iloc2: Initial values for the location parameters. By default,
          initial values are chosen internally using 'method.init'.
          Assigning values here will override the argument
          'method.init'. 

iscale1, iscale2: Initial values for the scale parameters. By default,
          initial values are chosen internally using 'method.init'.
          Assigning values here will override the argument
          'method.init'. 

method.init: An integer with value '1' or '2' which specifies the
          initialization method. If failure to converge occurs try the
          other value. 

    zero: An integer-valued vector specifying which linear/additive
          predictors are modelled as intercepts only. The default is
          none of them. If used, choose values from the set {1,2,3,4}. 

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

     The four-parameter bivariate logistic distribution  has a density
     that can be written as

 f(y1,y2;l1,s1,l2,s2) = 2 * exp[-(y1-l1)/s1 - (y1-l1)/s1] / [s1 * s2 * ( 1 + exp[-(y1-l1)/s1] + exp[-(y2-l2)/s2] )^3]

     where s1>0 s2>0 are the scale parameters, and l1 and l2 are the
     location parameters. Each of the two responses are unbounded,
     i.e., -Inf<y_j<Inf. The mean of Y1 is l1 etc. The fitted values
     are returned in a 2-column matrix. The cumulative distribution
     function is

 F(y1,y2;l1,s1,l2,s2) = 1 / (1 + exp[-(y1-l1)/s1] + exp[-(y2-l2)/s2])

     The marginal distribution of Y1 is

       P(Y1 <= y1) = F(y1;l1,s1) = 1 / (1 + exp[-(y1-l1)/s1]).


     By default, eta1=l1, eta2=log(s1), eta3=l2, eta4=log(s2) are the
     linear/additive predictors.

_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:

     This family function uses the BFGS quasi-Newton update formula for
     the working weight matrices.  Consequently the estimated
     variance-covariance matrix may be inaccurate or simply wrong! The
     standard errors must be therefore treated with caution; these are
     computed in functions such as 'vcov()' and 'summary()'.

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

     T. W. Yee

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

     Gumbel, E. J. (1961) Bivariate logistic distributions. _Journal of
     the American Statistical Association_, *56*, 335-349.

     Castillo, E., Hadi, A. S., Balakrishnan, N. Sarabia, J. S. (2005)
     _Extreme Value and Related Models with Applications in Engineering
     and Science_, Hoboken, N.J.: Wiley-Interscience.

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

     'logistic', 'rbilogis4'.

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

     ymat = rbilogis4(n <- 1000, loc1=5, loc2=7, scale2=exp(1))
     ## Not run: plot(ymat)
     fit = vglm(ymat ~ 1, fam=bilogistic4, trace=TRUE)
     coef(fit, matrix=TRUE)
     Coef(fit)
     fitted(fit)[1:4,]
     vcov(fit)
     weights(fit, type="w")[1:4,]
     summary(fit)

