paretoIV                package:VGAM                R Documentation

_P_a_r_e_t_o(_I_V/_I_I_I/_I_I) _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 three of the parameters of the Pareto(IV) distribution
     by maximum likelihood estimation. Some special cases of this
     distribution are also handled.

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

     paretoIV(location=0, lscale="loge", linequality="loge", lshape="loge",
              escale=list(), einequality=list(), eshape=list(),
              iscale=1, iinequality=1, ishape=NULL, method.init=1)
     paretoIII(location=0, lscale="loge", linequality="loge",
               escale=list(), einequality=list(),
               iscale=NULL, iinequality=NULL)
     paretoII(location=0, lscale="loge", lshape="loge",
              escale=list(), eshape=list(),
              iscale=NULL, ishape=NULL)

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

location: Location parameter, called a below. It is assumed known.

lscale, linequality, lshape: Parameter link functions for the scale
          parameter (called b below), inequality parameter (called g
          below), and  shape parameter (called s below). See 'Links'
          for more choices. A log link is the default for all because
          all these parameters are positive.

escale, einequality, eshape: List. Extra argument for each of the
          links. See 'earg' in 'Links' for general information.

iscale, iinequality, ishape: Initial values for the parameters. A
          'NULL' value means that it is obtained internally. If
          convergence failure occurs, use these arguments to input some
          alternative initial values.

method.init: Method of initialization for the shape parameter.
          Currently only values 1 and 2 are available. Try the other
          value if convergence failure occurs.

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

     The Pareto(IV) distribution, which is used in actuarial science,
     economics, finance and telecommunications, has a cumulative
     distribution function that can be written

                F(y) = 1 - [1 + ((y-a)/b)^(1/g)]^(-s)

     for y > a, b>0, g>0 and s>0. The a is called the _location_
     parameter, b the _scale_ parameter, g the _inequality_ parameter,
     and  s the _shape_ parameter.

     The location parameter is assumed known otherwise the Pareto(IV)
     distribution will not be a regular family.  This assumption is not
     too restrictive in modelling because in typical applications this
     parameter is known, e.g., in insurance and  reinsurance it is
     pre-defined by a contract and can be represented as a deductible
     or a retention level.

     The inequality parameter is so-called because of its
     interpretation in the economics context. If we choose a unit shape
     parameter value and a zero location parameter value then the
     inequality parameter is the Gini index of inequality, provided
     g<=1.

     The fitted values are currently 'NA' because I haven't worked out
     what the mean of Y is yet.

     There are a number of special cases of the Pareto(IV)
     distribution. These include the Pareto(I), Pareto(II),
     Pareto(III), and Burr family of distributions. Denoting
     PIV(a,b,g,s) as the Pareto(IV) distribution,  the Burr
     distribution Burr(b,g,s) is PIV(a=0,b,1/g,s), the Pareto(III)
     distribution PIII(a,b,g) is PIV(a,b,g,s=1), the Pareto(II)
     distribution PII(a,b,s) is PIV(a,b,g=1,s), and  the Pareto(I)
     distribution PI(b,s) is PIV(b,b,g=1,s). Thus the Burr distribution
     can be fitted using the 'nloge' link function and using the
     default 'location=0' argument. The Pareto(I) distribution can be
     fitted using 'pareto1' but there is a slight change in notation:
     s=k and b=alpha.

_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 'extra' slot of the fitted object has a component called
     '"location"' which stores the location parameter value(s).

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

     T. W. Yee

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

     Brazauskas, V. (2003) Information matrix for Pareto(IV), Burr, and
     related distributions. _Comm. Statist. Theory and Methods_ *32*,
     315-325.

     Arnold, B. C. (1983) _Pareto Distributions_. Fairland, Maryland:
     International Cooperative Publishing House.

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

     'ParetoIV', 'pareto1', 'gpd'.

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

     y = rparetoIV(n <- 2000, scale=exp(1), ineq=exp(-0.3), shape=exp(1))
     ## Not run: par(mfrow=c(2,1)); hist(y); hist(log(y)); 
     fit = vglm(y ~ 1, paretoIV, trace=TRUE)
     coef(fit, matrix=TRUE)
     Coef(fit)
     summary(fit)

