frechet                 package:VGAM                 R Documentation

_F_r_e_c_h_e_t _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 and 3-parameter
     Frechet distribution.

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

     frechet2(location=0, lscale="loge", lshape="loglog",
              escale = list(), eshape = list(),
              iscale=NULL, ishape=3, zero=NULL)
     frechet3(anchor=NULL, ldifference="loge",
              lscale="loge", lshape="loglog",
              edifference=list(), escale = list(), eshape = list(),
              ilocation=NULL, iscale=NULL, ishape=3, zero=NULL,
              effpos = .Machine$double.eps^0.75)

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

location: Numeric. Location parameter. It is called a below.

lscale, lshape: Link functions for the parameters. See 'Links' for more
          choices.

iscale, ishape: Initial value for the scale and shape parameters
          respectively.  Both parameters must be positive.  A 'NULL'
          means it is chosen internally.

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

    zero: An integer-valued vector specifying which linear/additive
          predictors are modelled as intercepts only.  The value must
          be from the set {1,2}, corresponding respectively to scale
          and shape.  By default all linear/additive predictors are
          modelled as a linear combination of the explanatory
          variables.

  anchor: An ``anchor'' point for estimating the location parameter.
          This must be a value no greater than 'min(y)' where 'y' is
          the response. The location parameter is A - D where A is the
          anchor, D is the ``difference'' (default is to make this
          positive). The default value of 'anchor' means 'min(y)' is
          chosen.

ldifference: Parameter link function for the difference D between the
          anchor point and the location parameter estimate. The default
          keeps this difference positive so that numerical problems are
          less likely to occur.

ilocation: Optional initial value for the location parameter. A good
          choice can speed up the convergence rate markedly. A 'NULL'
          means it is chosen internally.

  effpos: A small number, usually positive, to pass into the
          quasi-Newton updating function. A value greater than 'effpos'
          is considered effectively positive.

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

     The Frechet distribution has a density function that can be
     written

    f(y) = ((s*b) / (y-a)^2) * exp[-(b/(y-a))^s] * [b/(y-a)]^(s-1)

     for y>a and scale parameter b>0. The positive shape parameter is
     s. The cumulative distribution function is 

                      F(y) =  exp[-(b/(y-a))^s].

     The mean of Y is a + b*gamma(1-1/s) for s>1 (these are returned as
     the fitted values). Note that the 'loglog' link ensures s>1. The
     variance of Y is b^2 * [gamma(1-2/s) - gamma(1-1/s)^2] for s>2.

     'frechet2' has a known whereas 'frechet3' estimates it. 
     Estimating a well requires a lot of data and a good choice of
     'ilocation' will help speed up convergence. For 'frechet3' the
     default linear/additive predictors are log(D),  log(b), 
     log(log(s)), respectively. It would be great if the first
     linear/additive predictor was a direct function of the location
     parameter, but this can run the risk that the estimate is out of
     range (i.e., greater than 'min(y)').

_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:

     Convergence for 'frechet3' can be very slow, especially if the
     initial value for the location parameter is poor. Setting
     something like 'maxit=200, trace=TRUE' is a good idea.

_N_o_t_e:

     These family functions use the BFGS quasi-Newton update formula
     for the working weight matrices.  Consequently the estimated
     variance-covariance matrix may be inaccurate or simply wrong! The
     standard errors must be therefore treated with caution; these are
     computed in functions such as 'vcov()' and 'summary()'.

     If 'fit' is a 'frechet3' fit then 'fit@extra$location' is the
     final estimate of the location parameter, and
     'fit@extra$LHSanchor' is the anchor point.

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

     T. W. Yee

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

     Castillo, E., Hadi, A. S., Balakrishnan, N. Sarabia, J. S. (2005)
     _Extreme Value and Related Models with Applications in Engineering
     and Science_, Hoboken, N.J.: Wiley-Interscience.

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

     'rfrechet'.

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

     y = rfrechet(n <- 1000, shape=exp(exp(0)))
     ## Not run: hist(y)
     fit2 = vglm(y ~ 1, frechet2, trace=TRUE, maxit=155)
     fit3 = vglm(y ~ 1, frechet3(ilocation=0), trace=TRUE, maxit=155)
     coef(fit3, matrix=TRUE)
     Coef(fit3)
     fitted(fit3)[1:5,]
     mean(y)
     weights(fit3, type="w")[1:5,]
     vcov(fit3)   # caution needed!
     fit3@extra$location[1:3]  # Estimate of the location parameter
     fit3@extra$LHSanchor  # Anchor point
     min(y)

