GeneralisedLambdaDistribution {gld}R Documentation

The Generalised Lambda Distribution

Description

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.

Usage

dgl(x, lambda1 = 0, lambda2 = NULL, lambda3 = NULL, lambda4 = NULL, 
  param = "fmkl", lambda5 = NULL, inverse.eps = 1e-08, 
  max.iterations = 500)
qdgl(p, lambda1, lambda2 = NULL, lambda3 = NULL, lambda4 = NULL, 
  param = "fmkl", lambda5 = NULL)
pgl(q, lambda1 = 0, lambda2 = NULL, lambda3 = NULL, lambda4 = NULL, 
  param = "fmkl", lambda5 = NULL, inverse.eps = 1e-08, 
  max.iterations = 500)
qgl(p, lambda1, lambda2 = NULL, lambda3 = NULL, lambda4 = NULL,
  param = "fmkl", lambda5 = NULL)
rgl(n, lambda1=0, lambda2 = NULL, lambda3 = NULL, lambda4 = NULL,
  param = "fmkl", lambda5 = NULL)

Arguments

x,q vector of quantiles.
p vector of probabilities.
n number of observations.
lambda1 This can be either a single numeric value or a vector.
If it is a vector, it must be of length 4 for parameterisations fmkl or rs and of length 5 for parameterisation fm5. If it is a vector, it gives all the parameters of the generalised lambda distribution (see below for details) and the other lambda arguments must be left as NULL.
If it is a a single value, it is lambda 1, the location parameter of the distribution and the other parameters are given by the following arguments
Note that the numbering of the lambda parameters for the fmkl parameterisation 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
lambda5 lambda 5 - a skewing parameter, in the fm5 parameterisation
param choose parameterisation: fmkl uses Freimer, Mudholkar, Kollia and Lin (1988) (default). rs uses Ramberg and Schmeiser (1974) fm5 uses the 5 parameter version of the FMKL parameterisation (paper to appear)
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

Details

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. The gld package implements three parameterisations of the distribution. The default parameterisation (the FMKL) 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.

A second parameterisation, the RS, 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.

A third parameterisation, the FM5, chosen by setting param="fm5" adds an additional skewing parameter to the FMKL parameterisation. This uses the same approach as that used by Karian & Dudwicz (2000) OR IS IT GILCHRIST?? for the RS parameterisation. The quantile function is

F inverse (u) = lambda1 + ( (1-lambda5)(u^lambda3-1)/lambda3 - (1+lambda5)((1-u)^lambda4-1)/lambda4 ) / lambda 2

for lambda2 >0 and -1 <= lambda5 <= 1.

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.

The functions qdgl.fmkl, qdgl.rs, qdgl.fm5, qgl.fmkl, qgl.rs and qgl.fm5 from versions 1.5 and earlier of the gld package have been renamed (and hidden) to .qdgl.fmkl, .qdgl.rs, ..qdgl.fm5, .qgl.fmkl, .qgl.rs and .qgl.fm5 respectively. See the code for more details

Value

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.

Author(s)

Robert King, robert.king@newcastle.edu.au, http://tolstoy.newcastle.edu.au/~rking/

References

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.

Gilchrist, Warren G. (2000), Statistical Modelling with Quantile Functions, Chapman & Hall

Karian, Z.A., 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.

Karian, Zaven A. and Dudewicz, Edward J. (2000), Fitting statistical distributions: the Generalized Lambda Distribution and Generalized Bootstrap methods, Chapman & Hall

Ramberg, J. S. & Schmeiser, B. W. (1974), An approximate method for generating asymmetric random variables, Communications of the ACM 17, 78–82.

http://tolstoy.newcastle.edu.au/~rking/gld/

Examples

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

[Package gld version 1.8.3 Index]