GeneralisedLambdaDistribution      package:gld      R Documentation

_T_h_e _G_e_n_e_r_a_l_i_s_e_d _L_a_m_b_d_a _D_i_s_t_r_i_b_u_t_i_o_n

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

     Density, quantile density, distribution function, quantile 
     function and random generation for the generalised lambda
     distribution  (also known as the asymmetric lambda, or Tukey
     lambda).  Works for  both the 'fmkl' (recommended) and 'rs'
     parameterisations.

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

     dgl(x, lambda1, lambda2, lambda3, lambda4,
     param="fmkl",inverse.eps=1e-8,max.iterations=500)
     qdgl(p, lambda1, lambda2, lambda3, lambda4, param="fmkl")
     qdgl.fmkl(p, lambda1, lambda2, lambda3, lambda4)
     qdgl.rs(p, lambda1, lambda2, lambda3, lambda4)
     pgl(q, lambda1, lambda2, lambda3, lambda4,
     param="fmkl",inverse.eps=1e-8,max.iterations=500)
     qgl(p, lambda1, lambda2, lambda3, lambda4, param="fmkl")
     qgl.fmkl(p, lambda1, lambda2, lambda3, lambda4)
     qgl.rs(p, lambda1, lambda2, lambda3, lambda4)
     rgl(n, lambda1, lambda2, lambda3, lambda4, param="fmkl")

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

     x,q: vector of quantiles.

       p: vector of probabilities.

       n: number of observations.

 lambda1: lambda 1 - location parameter _Note that the numbering of the
          lambda parameters  is different to that used by Freimer,
          Mudholkar, Kollia and Lin._ 

 lambda2: lambda 2 - scale parameter

 lambda3: lambda 3 - first shape parameter

 lambda4: lambda 4 - second shape parameter

   param: choose parameterisation:  'fmkl' uses _Freimer, Mudholkar,
          Kollia and Lin (1988)_ (default). 'rs' uses _Ramberg and
          Schmeiser (1974)_

inverse.eps: Accuracy of calculation for the numerical determination of
           F(x), defaults to 1e-8

max.iterations: Maximum number of iterations in the numerical 
          determination of F(x), defaults to 500

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

     The generalised lambda distribution, also known as the asymmetric
     lambda,  or Tukey lambda distribution, is a distribution with a
     wide range of shapes.   The distribution is defined by its
     quantile function, the inverse of the distribution function. 
     There are two parameterisations of the distribution.   The default
     parameterisation is that due to _Freimer Mudholkar, Kollia  and
     Lin (1988)_ (see references below), with a quantile function:

 F inverse (u) = lambda1 + ( (u^lambda3 -1)/lambda3 - ((1-u)^lambda4    -1)/lambda4 ) / lambda 2

     for lambda2 >0.

     The alternative parameterisation, chosen by setting 'param="rs"'
     is that due to _Ramberg and Schmeiser (1974)_, with the quantile
     function:

 F inverse (u) = lambda1 + ( u^lambda3 - (1-u)^lambda4 ) /  lambda 2


     This parameterisation has a complex series of rules determining
     which values  of the parameters produce valid statistical
     distributions.  See  gl.check.lambda for details.

     The distribution is defined by its quantile function and its
     distribution and density functions do not exist in closed form. 
     Accordingly, the results  from 'pgl' and 'dgl' are the result of a
     numerical solutions to equations, using the Newton-Raphson method.
      Since the quantile density function, f(F^{-1}(u)), does exist, an
     additional function, 'qdgl', computes this.

     'qdgl.fmkl' and 'qdgl.rs' are versions of 'qdgl' that assume the 
     FMKL and RS parameterisations, respectively.

     'qgl.fmkl' and 'qgl.rs' are versions of 'qgl' that assume the 
     FMKL and RS parameterisations, respectively.

_V_a_l_u_e:

     'dgl' gives the density (based on the quantile density and a 
     numerical solution to F inv (u)=x), 

     'qdgl' gives the quantile density,

     'pgl' gives the distribution function (based on a numerical
     solution to F inv (u)=x),

     'qgl' gives the quantile function, and

     'rgl' generates random deviates.

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

     Robert King, robert.king@newcastle.edu.au, <URL:
     http://maths.newcastle.edu.au/~rking/>

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

     Freimer, M., Mudholkar, G. S., Kollia, G. & Lin, C. T. (1988),  _A
     study of the generalized tukey lambda family_, Communications  in
     Statistics - Theory and Methods *17*, 3547-3567.

     Karian, Z.E., Dudewicz, E.J., and McDonald, P. (1996), _The
     extended  generalized lambda distribution system for fitting
     distributions to data:  history, completion of theory, tables,
     applications, the ``Final Word''  on Moment fits_, Communications
     in Statistics - Simulation and Computation  *25*, 611-642.

     Ramberg, J. S. & Schmeiser, B. W. (1974), _An approximate method
     for generating asymmetric random variables_, Communications of the
     ACM *17*,  78-82.

     <URL: http://maths.newcastle.edu.au/~rking/gld/>

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

     qgl(seq(0,1,0.02),0,1,0.123,-4.3)
     pgl(seq(-2,2,0.2),0,1,-.1,-.2,param="fmkl",inverse.eps=.Machine$double.eps)
     # calculate the probabilities more accurately than normal

