cauchy                 package:VGAM                 R Documentation

_C_a_u_c_h_y _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:

     Estimates either the location parameter or both the location and
     scale parameters of the Cauchy distribution by maximum likelihood
     estimation.

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

     cauchy(llocation="identity", lscale="loge", elocation=list(),
            escale=list(), ilocation=NULL, iscale=NULL,
            iprobs = seq(0.2, 0.8, by=0.2),
            method.init=1, nsimEIM=NULL, zero=2)
     cauchy1(scale.arg=1, llocation="identity",
             elocation=list(), ilocation=NULL, method.init=1)

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

llocation, lscale: Parameter link functions for the location parameter
          a and the scale parameter b. See 'Links' for more choices.

elocation, escale: List. Extra argument for each link. See 'earg' in
          'Links' for general information.

ilocation, iscale: Optional initial value for a and b. By default, an
          initial value is chosen internally for each.

method.init: Integer, either 1 or 2 or 3. Initial method, three
          algorithms are implemented. The user should try all possible
          values to help avoid converging to a local solution. Also,
          choose the another value if convergence fails, or use
          'ilocation' and/or 'iscale'.

  iprobs: Probabilities used to find the respective sample quantiles;
          used to compute 'iscale'.

zero, nsimEIM: See 'CommonVGAMffArguments' for more information.

scale.arg: Known (positive) scale parameter, called b below.

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

     The Cauchy distribution has density function 

             f(y;a,b) = 1 / [pi * b * [1 + ((y-a)/b)^2]]

     where y and a are real and finite, and b>0. The distribution is
     symmetric about a and has a heavy tail. Its median and mode are a
     but the mean does not exist. The fitted values are the estimates
     of a. Fisher scoring is the default but if 'nsimEIM' is specified
     then Fisher scoring with simulation is used.

     If the scale parameter is known ('cauchy1') then there may be
     multiple local maximum likelihood solutions for the location
     parameter. However, if both location and scale parameters are to
     be estimated ('cauchy') then there is a unique maximum likelihood
     solution provided n > 2 and less than half the data are located at
     any one point.

_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:

     It is well-known that the Cauchy distribution may have local
     maxima in its likelihood function; make full use of 'method.init',
     'ilocation', 'iscale' etc.

_N_o_t_e:

     Good initial values are needed. By default these 'VGAM' family
     functions search for a starting value for a on a grid. It also
     pays to select a wide range of initial values via the 'ilocation'
     and/or 'iscale' and/or 'method.init' arguments.

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

     T. W. Yee

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

     Evans, M., Hastings, N. and Peacock, B. (2000) _Statistical
     Distributions_, New York: Wiley-Interscience, Third edition.

     Barnett, V. D. (1966) Evaluation of the maximum-likehood estimator
     where the likelihood equation has multiple roots. _Biometrika_,
     *53*, 151-165.

     Copas, J. B. (1975) On the unimodality of the likelihood for the
     Cauchy distribution. _Biometrika_, *62*, 701-704.

     Efron, B. and Hinkley, D. V. (1978) Assessing the accuracy of the
     maximum likelihood estimator: Observed versus expected Fisher
     information. _Biometrika_, *65*, 457-481.

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

     'Cauchy', 'cauchit'.

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

     # Both location and scale parameters unknown
     x = runif(n <- 1000)
     y = rcauchy(n, loc=exp(1+0.5*x), scale=exp(1))
     fit = vglm(y ~ x, cauchy(lloc="loge"), trace=TRUE)
     coef(fit, matrix=TRUE)
     head(fitted(fit))  # Location estimates
     summary(fit)

     # Location parameter unknown
     set.seed(123)
     x = runif(n <- 500)
     y = rcauchy(n, loc=1+5*x, scale=0.4)
     fit = vglm(y ~ x, cauchy1(scale=0.4), trace=TRUE, crit="c")
     coef(fit, matrix=TRUE)

