hzeta                  package:VGAM                  R Documentation

_H_a_i_g_h_t'_s _Z_e_t_a _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:

     Estimating the parameter of Haight's Zeta function.

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

     hzeta(link = "loglog", init.alpha = NULL)

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

    link: Parameter link function for the parameter. See 'Links' for
          more choices. Here, a log-log link keeps the parameter
          greater than one, meaning the mean is finite.

init.alpha: Optional initial value for the (positive) parameter.  The
          default is to obtain an initial value internally. Use this
          argument if the default fails.

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

     The probability function is

              f(y) = (2y-1)^(-alpha) - (2y+1)^(-alpha),

     where the parameter alpha>0 and y=1,2,.... The function 'dhzeta'
     computes this probability function. The mean of Y, which is
     returned as fitted values, is (1-2^(-alpha))*zeta(alpha) provided
     alpha > 1, where zeta is Riemann's zeta function. The mean is a
     decreasing function of alpha. The mean is infinite if alpha <= 1,
     and the variance is infinite if alpha <= 2.

_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'.

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

     T. W. Yee

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

     Page 470 of Johnson N. L., Kotz S. and Kemp, A. W. (1993)
     _Univariate Discrete Distributions_, 2nd edition, Volume 2, New
     York: Wiley.

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

     'Hzeta', 'zeta', 'zetaff', 'loglog'.

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

     alpha = exp(0.1)  # The parameter
     y = rhzeta(n=400, alpha) # Generate some hzeta random variates
     fit = vglm(y ~ 1, hzeta, trace = TRUE, crit="c")
     coef(fit, matrix=TRUE)
     Coef(fit)  # Useful for intercept-only models; should be same as alpha
     fitted(fit)[1:4,]

