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, 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")

Arguments

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

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. 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.

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://maths.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.

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.

http://maths.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 1.6 Index]