gammahyp                package:VGAM                R Documentation

_G_a_m_m_a _H_y_p_e_r_b_o_l_a _B_i_v_a_r_i_a_t_e _D_i_s_t_r_i_b_u_t_i_o_n

_D_e_s_c_r_i_p_t_i_o_n:

     Estimate the parameter of a gamma hyperbola bivariate distribution
     using maximum likelihood estimation.

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

     gammahyp(ltheta="loge", itheta=NULL, expected=FALSE)

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

  ltheta: Link function applied to the (positive) parameter theta. See
          'Links' for more choices.

  itheta: Initial value for the parameter. The default is to estimate
          it internally.

expected: Logical. 'FALSE' means the Newton-Raphson (using the observed
          information matrix) algorithm, otherwise the expected
          information matrix is used (Fisher scoring algorithm).

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

     The joint probability density function is given by

       f(y1,y2) = exp( -exp(-theta) * y1 / theta - theta * y2)

     for theta > 0, y1 > 0, y2 > 1. The random variables Y1  and Y2 are
     independent. The marginal distribution of Y1 is an exponential
     distribution with rate parameter exp(-theta)/theta. The marginal
     distribution of Y2 is an exponential distribution that has been
     shifted to the right by 1 and with rate parameter theta. The
     fitted values are stored in a two-column matrix with the marginal
     means, which are theta * exp(theta) and 1 + 1/theta.

     The default algorithm is Newton-Raphson because Fisher scoring
     tends to be much slower for this distribution.

_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.

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

     T. W. Yee

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

     Reid, N. (2003) Asymptotics and the theory of inference.  _Annals
     of Statistics_, *31*, 1695-1731.

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

     'exponential'.

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

     x  = runif(n <- 1000)
     theta = exp(-2+x)
     y1 = rexp(n, rate=exp(-theta)/theta)
     y2 = 1 + rexp(n, rate=theta)
     fit = vglm(cbind(y1,y2) ~ x, fam=gammahyp(expected=TRUE), trace=TRUE)
     fit = vglm(cbind(y1,y2) ~ x, fam=gammahyp, trace=TRUE, crit="coef")
     coef(fit, matrix=TRUE)
     Coef(fit)
     fitted(fit)[1:4,]
     summary(fit)

