tikuv                  package:VGAM                  R Documentation

_S_h_o_r_t-_t_a_i_l_e_d _S_y_m_m_e_t_r_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:

     Fits the short-tailed symmetric distribution of Tiku and Vaughan
     (1999).

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

     tikuv(d, lmean="identity", lsigma="loge", isigma=NULL, zero=2)

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

       d: The d parameter. It must be a single numeric value less than
          2. Then h=2-d>0 is another parameter.

lmean, lsigma: Link functions for the mean and standard deviation
          parameters of the usual univariate normal distribution (see
          *Details* below). They are mu and sigma respectively. See
          'Links' for more choices.

  isigma: Optional initial value for sigma. A 'NULL' means a value is
          computed internally.

    zero: An integer-valued vector specifying which linear/additive
          predictors are modelled as intercepts only. The values must
          be from the set {1,2} corresponding respectively to mu,
          sigma. If 'zero=NULL' then all linear/additive predictors are
          modelled as a linear combination of the explanatory
          variables. For many data sets having 'zero=2' is a good idea.

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

     The short-tailed symmetric distribution of Tiku and Vaughan (1999)
     has a probability density function that can be written

 f(y) = (K/(sqrt(2*pi)*sigma)) * [1 + (1/(2*h)) * ((y-mu)/sigma)^2]^2 * exp( -0.5 * (y-mu)^2/  sigma^2)

     where h=2-d>0, K is a function of h, -Inf < y < Inf, sigma > 0.
     The mean of Y is E(Y) = mu and this is returned as the fitted
     values.

_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'.

_W_a_r_n_i_n_g:

     Under- or over-flow may occur if the data is ill-conditioned,
     e.g., when d is very close to 2 or approaches '-Inf'.

_N_o_t_e:

     The density function is the product of a univariate normal density
     and a polynomial in the response y. The distribution is bimodal if
     d>0, else is unimodal. A normal distribution arises as the limit
     as d approaches -Inf, i.e., as h approaches Inf. Fisher scoring is
     implemented. After fitting the value of 'd' is stored as
     '@misc\$d'.

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

     Thomas W. Yee

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

     Akkaya, A. D. and Tiku, M. L. (2006) Short-tailed distributions
     and inliers. _Test_, *15*(2), in press.

     Tiku, M. L. and Vaughan, D. C. (1999) A family of short-tailed
     symmetric distributions. _Technical report, McMaster University,
     Canada_.

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

     'dtikuv', 'normal1'.

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

     m = 1.0; sigma = exp(0.5)
     sy = sort(y <- rtikuv(n=1000, d=1, m=m, s=sigma))
     fit = vglm(y ~ 1, fam=tikuv(d=1), trace=TRUE)
     coef(fit, mat=TRUE)
     (Cfit = Coef(fit))
     mean(y)
     ## Not run: 
     hist(y, prob=TRUE)
     lines(sy, dtikuv(sy, d=1, m=Cfit[1], s=Cfit[2]), col="red")
     ## End(Not run)

