genpoisson               package:VGAM               R Documentation

_G_e_n_e_r_a_l_i_z_e_d _P_o_i_s_s_o_n _d_i_s_t_r_i_b_u_t_i_o_n

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

     Estimation of the two parameters of a generalized Poisson
     distribution.

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

     genpoisson(llambda="elogit", ltheta="loge",
                elambda=if(llambda=="elogit") list(min=-1,max=1) else list(),
                etheta=list(), ilambda=NULL, itheta=NULL,
                use.approx=TRUE, method.init=1, zero=1)

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

llambda, ltheta: Parameter link functions for lambda and theta. See
          'Links' for more choices. The lambda parameter lies at least
          within the interval [-1,1]; see below for more details. The
          theta parameter is positive, therefore the default is the log
          link.

elambda, etheta: List. Extra argument for each of the links. See 'earg'
          in 'Links' for general information.

ilambda, itheta: Optional initial values for lambda and theta. The
          default is to choose values internally.

use.approx: Logical. If 'TRUE' then an approximation to the expected
          information matrix is used, otherwise Newton-Raphson is used.

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

    zero: An integer vector, containing the value 1 or 2. If so, lambda
          or theta respectively are modelled as an intercept only. If
          set to 'NULL' then both linear/additive predictors are
          modelled as functions of the explanatory variables.

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

     The generalized Poisson distribution has density

     f(y) = theta(theta+lambda y)^{y-1} exp(-theta-lambda y) / y!


     for theta > 0 and y = 0,1,2,.... Now max(-1,-theta/m) <= lambda <=
     1 where m (>= 4) is the greatest positive integer satisfying theta
     + mlambda > 0 when lambda < 0 [and then P(Y=y)=0 for y > m]. Note
     the complicated support for this distribution means, for some data
     sets, the default link for 'llambda' is not always appropriate.

     An ordinary Poisson distribution corresponds to lambda=0. The mean
     (returned as the fitted values) is  E(Y) = theta / (1 - lambda)
     and the variance is theta / (1 - lambda)^3.

     For more information see Consul and Famoye (2006) for a summary
     and Consul (1989) for full details.

_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:

     This distribution is useful for dispersion modelling. Convergence
     problems may occur when 'lambda' is very close to 0 or 1.

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

     T. W. Yee

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

     Consul, P. C. and Famoye, F. (2006) _Lagrangian Probability
     Distributions_, Boston: Birkhauser.

     Jorgensen, B. (1997) _The Theory of Dispersion Models_. London:
     Chapman & Hall

     Consul, P. C. (1989) _Generalized Poisson Distributions:
     Properties and Applications_. New York: Marcel Dekker.

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

     'poissonff'.

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

     n = 200
     x = runif(n)
     y = rpois(n, lam=exp(2-x))
     fit  = vglm(y ~ x, genpoisson(zero=1), trace=TRUE)
     coef(fit, matrix=TRUE)
     summary(fit)

