weibull                 package:VGAM                 R Documentation

_W_e_i_b_u_l_l _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 2-parameter Weibull
     distribution. No observations should be censored.

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

     weibull(lshape = "loge", lscale = "loge", 
             eshape = list(), escale = list(),
             ishape = NULL, iscale = NULL,
             nrfs = 1, imethod=1, zero = 2)

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

lshape, lscale: Parameter link functions applied to the  (positive)
          shape parameter (called a below) and (positive) scale
          parameter (called b below). See 'Links' for more choices.

eshape, escale: Extra argument for the respective links. See 'earg' in
          'Links' for general information.

ishape, iscale: Optional initial values for the shape and scale
          parameters.

    nrfs: Currently this argument is ignored. Numeric, of length one,
          with value in [0,1]. Weighting factor between Newton-Raphson
          and Fisher scoring. The value 0 means pure Newton-Raphson,
          while 1 means pure Fisher scoring. The default value uses a
          mixture of the two algorithms, and retaining
          positive-definite working weights.

 imethod: Initialization method used if there are censored
          observations. Currently only the values 1 and 2 are allowed. 

    zero: An integer specifying which linear/additive predictor is to
          be modelled as an intercept only.  The value must be from the
          set {1,2}, which correspond to the shape and scale parameters
          respectively. Setting 'zero=NULL' means none of them.

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

     The Weibull density for a response Y is 

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

     for a > 0, b > 0, y > 0. The cumulative distribution function is 

                    F(y;a,b) = 1 - exp(-(y/b)^a).

     The mean of Y is b * gamma(1+ 1/a) (returned as the fitted
     values), and the mode is at b * (1- 1/a)^(1/a) when a>1. The
     density is unbounded for a<1. The kth moment about the origin is
     E(Y^k) = b^k * gamma(1+ k/a). The hazard function is a * t^(a-1) /
     b^a.

     This 'VGAM' family function currently does not handle  censored
     data. Fisher scoring is used to estimate the two parameters.
     Although the Fisher information matrices used here are valid in
     all regions of the parameter space, the regularity conditions for
     maximum likelihood estimation are satisfied only if a>2 (according
     to Kleiber and Kotz (2003)). If this is violated then a warning
     message is issued. One can enforce a>2 by choosing 'lshape =
     "logoff"' and 'eshape=list(offset=-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'.

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

     This function is under development to handle other censoring
     situations. The version of this function which will handle
     censored data will be called 'cenweibull()'. It is currently being
     written and will use 'Surv' as input.  It should be released in
     later versions of 'VGAM'.

     If the shape parameter is less than two then misleading inference
     may result, e.g., in the 'summary' and 'vcov' of the object.

_N_o_t_e:

     Successful convergence depends on having reasonably good initial
     values. If the initial values chosen by this function are not
     good, make use the two initial value arguments.

     The Weibull distribution is often an alternative to the lognormal
     distribution.  The inverse Weibull distribution, which is that of
     1/Y where Y has a Weibull(a,b) distribution, is known as the
     log-Gompertz distribution.

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

     T. W. Yee

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

     Kleiber, C. and Kotz, S. (2003) _Statistical Size Distributions in
     Economics and Actuarial Sciences_, Hoboken, NJ:
     Wiley-Interscience.

     Johnson, N. L. and Kotz, S. and Balakrishnan, N. (1994)
     _Continuous Univariate Distributions_, 2nd edition, Volume 1, New
     York: Wiley.

     Gupta, R. D. and Kundu, D. (2006) On the comparison of Fisher
     information of the Weibull and GE distributions, _Journal of
     Statistical Planning and Inference_, *136*, 3130-3144.

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

     'dweibull', 'gev', 'lognormal', 'expexp'.

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

     # Complete data
     x = runif(n <- 1000)
     y = rweibull(n, shape=exp(1+x), scale = exp(-0.5))
     fit = vglm(y ~ x, weibull, trace=TRUE)
     coef(fit, mat=TRUE)
     vcov(fit)
     summary(fit)

