lino                  package:VGAM                  R Documentation

_G_e_n_e_r_a_l_i_z_e_d _B_e_t_a _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:

     Maximum likelihood estimation of the 3-parameter  generalized beta
     distribution as proposed by Libby and Novick (1982).

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

     lino(lshape1="loge", lshape2="loge", llambda="loge",
          eshape1=list(), eshape2=list(), elambda=list(),
          ishape1=NULL, ishape2=NULL, ilambda=1, zero=NULL)

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

lshape1, lshape2: Parameter link functions applied to the two
          (positive) shape parameters a and b. See 'Links' for more
          choices.

 llambda: Parameter link function applied to the  parameter lambda. See
          'Links' for more choices.

eshape1, eshape2, elambda: List. Extra argument for each of the links.
          See 'earg' in 'Links' for general information.

ishape1, ishape2, ilambda: Initial values for the parameters. A 'NULL'
          value means one is computed internally. The argument
          'ilambda' must be numeric, and the default corresponds to a
          standard beta distribution.

    zero: An integer-valued vector specifying which linear/additive
          predictors are modelled as intercepts only. Here, the values
          must be from the set {1,2,3} which correspond to a, b,
          lambda, respectively.

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

     Proposed by Libby and Novick (1982), this distribution has density

 f(y;a,b,lambda) = lambda^a y^(a-1) (1-y)^(b-1) / [B(a,b) (1 - (1-lambda)*y)^(a+b)]

     for a > 0, b > 0, lambda > 0, 0 < y < 1. Here B is the beta
     function (see 'beta'). The mean is a complicated function
     involving the Gauss hypergeometric function. If X has a 'lino'
     distribution with parameters 'shape1', 'shape2', 'lambda', then Y
     = lambda*X / (1 - (1-lambda)*X) has a standard beta distribution
     with parameters 'shape1', 'shape2'.

     Since log(lambda)=0 corresponds to the standard beta distribution,
     a 'summary' of the fitted model performs a t-test for whether the
     data belongs to a standard beta distribution (provided the 'loge'
     link for lambda is used; this is the default).

_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 fitted values, which is usually the mean, have not been
     implemented yet and consequently are 'NA's.

     Although Fisher scoring is used, the working weight matrices are
     positive-definite only in a certain region of the parameter space.
     Problems with this indicate poor initial values or an
     ill-conditioned model or insufficient data etc.

     This model is can be difficult to fit. A reasonably good value of
     'ilambda' seems to be needed so if the self-starting initial
     values fail, try experimenting with the initial value arguments.
     Experience suggests 'ilambda' is better a little larger, rather
     than smaller, compared to the true value.

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

     T. W. Yee

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

     Libby, D. L. and Novick, M. R. (1982) Multivariate generalized
     beta distributions with applications to utility assessment.
     _Journal of Educational Statistics_, *7*, 271-294.

     Gupta, A. K. and Nadarajah, S. (2004) _Handbook of Beta
     Distribution and Its Applications_, NY: Marcel Dekker, Inc.

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

     'Lino', 'genbetaII'.

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

     y = rbeta(n=1000, exp(0.5), exp(1)) # Standard beta distribution
     fit = vglm(y ~ 1, lino, trace=TRUE)
     coef(fit, mat=TRUE)
     Coef(fit)
     fitted(fit)[1:4]
     summary(fit)

     # Nonstandard beta distribution
     y = rlino(n=1000, shape1=2, shape2=3, lambda=exp(1))
     fit = vglm(y ~ 1, lino(lshape1=identity, lshape2=identity, ilambda=10))
     coef(fit, mat=TRUE)

