pareto1                 package:VGAM                 R Documentation

_P_a_r_e_t_o _a_n_d _T_r_u_n_c_a_t_e_d _P_a_r_e_t_o _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_s

_D_e_s_c_r_i_p_t_i_o_n:

     Estimates one of the parameters of the Pareto(I) distribution by
     maximum likelihood estimation. Also includes the upper truncated
     Pareto(I) distribution.

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

     pareto1(lshape = "loge", earg=list(), location=NULL)
     tpareto1(lower, upper, lshape = "loge", earg=list(), ishape=NULL,
              method.init=1)

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

  lshape: Parameter link function applied to the parameter k. See
          'Links' for more choices. A log link is the default because k
          is positive.

    earg: List. Extra argument for the link. See 'earg' in 'Links' for
          general information.

lower, upper: Numeric. Lower and upper limits for the truncated Pareto
          distribution. Each must be positive and of length 1. They are
          called alpha and U below.

  ishape: Numeric. Optional initial value for the shape parameter. A
          'NULL' means a value is obtained internally. If failure to
          converge occurs try specifying a value, e.g., 1 or 2.

location: Numeric. The parameter alpha below. If the user inputs a
          number then it is assumed known with this value. The default
          means it is estimated by maximum likelihood estimation, which
          means 'min(y)' where 'y' is the response vector.

method.init: An integer with value '1' or '2' which specifies the
          initialization method. If failure to converge occurs try the
          other value, or else specify a value for 'ishape'.

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

     A random variable Y has a Pareto distribution if

                           P[Y>y] = C / y^k

     for some positive k and C. This model is important in many
     applications due to the power law probability tail, especially for
     large values of y.

     The Pareto distribution, which is used a lot in economics, has a
     probability density function that can be written

                     f(y) = k * alpha^k / y^(k+1)

     for 0< alpha < y and k>0. The alpha is known as the location
     parameter, and  k is known as the shape parameter. The mean of Y
     is alpha*k/(k-1) provided k>1. Its variance is alpha^2 k /((k-1)^2
     (k-2)) provided k>2.

     The upper truncated Pareto distribution has a probability density
     function that can be written

            f(y) = k * alpha^k / [y^(k+1) (1-(alpha/U)^k)]

     for 0< alpha < y < U < Inf and k>0. Possibly, better names for k
     are  the _index_ and _tail_ parameters. Here, alpha and U are
     known. The mean of Y is k * lower^k * (U^(1-k)-alpha^(1-k)) /
     ((1-k) * (1-(alpha/U)^k)).

_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:

     The usual or unbounded Pareto distribution has two parameters
     (called alpha and k here) but the family function 'pareto1'
     estimates only k using iteratively reweighted least squares. The
     MLE of the alpha parameter lies on the boundary and is 'min(y)'
     where 'y' is the response. Consequently, using the default
     argument values, the standard errors are incorrect when one does a
     'summary' on the fitted object.  If the user inputs a value for
     'alpha' then it is assumed known with this value and then
     'summary' on the fitted object should be correct.  Numerical
     problems may occur for small k, e.g., k < 1.

_N_o_t_e:

     Outside of economics, the Pareto distribution is known as the
     Bradford distribution.

     For 'pareto1', if the estimate of k is less than or equal to unity
     then the fitted values will be 'NA's. Also, 'pareto1' fits the
     Pareto(I) distribution. See 'paretoIV' for the more general
     Pareto(IV/III/II) distributions, but there is a slight change in
     notation: s=k and b=alpha.

     In some applications the Pareto law is truncated by a  natural
     upper bound on the probability tail. The upper truncated Pareto
     distribution has three parameters (called alpha, U and k here) but
     the family function 'tpareto' estimates only k. With known lower
     and upper limits, the ML estimator of k has the usual properties
     of MLEs.  Aban (2006) discusses other inferential details.

_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.

     Aban, I. B., Meerschaert, M. M. and Panorska, A. K. (2006)
     Parameter estimation for the truncated Pareto distribution,
     _Journal of the American Statistical Association_, *101*(473),
     270-277.

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

     'Pareto', 'Tpareto', 'paretoIV', 'gpd'.

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

     alpha = 2; k = exp(3)
     y = rpareto(n=1000, location=alpha, shape=k)
     fit = vglm(y ~ 1, pareto1, trace=TRUE)
     fit@extra   # The estimate of alpha is here
     fitted(fit)[1:5]
     mean(y)
     coef(fit, matrix=TRUE)
     summary(fit)     # Standard errors are incorrect!!

     # Here, alpha is assumed known
     fit2 = vglm(y ~ 1, pareto1(location=alpha), trace=TRUE, crit="c")
     fit2@extra   # alpha stored here
     fitted(fit2)[1:5]
     mean(y)
     coef(fit2, matrix=TRUE)
     summary(fit2)    # Standard errors are ok

     # Upper truncated Pareto distribution
     lower = 2; upper = 8; k = exp(2)
     y = rtpareto(n=100, lower=lower, upper=upper, shape=k)
     fit3 = vglm(y ~ 1, tpareto1(lower, upper), trace=TRUE, cri="c")
     coef(fit3, matrix=TRUE)
     c(fit3@misc$lower, fit3@misc$upper)

