The Generalised Lambda Distribution Family
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" and "rs"
parameterisations. These functions originate from the gld
library by Robert King and they are modified in this package
to allow greater functionality and adaptability to new fitting
methods. It does not give an error message for invalid distributions
but will return NAs instead. To allow comparability with the pkg(gld)
package, this package uses the same notation and description as those
written by Robert King.
dgl(x, lambda1 = 0, lambda2 = NULL, lambda3 = NULL, lambda4 = NULL,
param = "fmkl", inverse.eps = 1e-08,
max.iterations = 500)
pgl(q, lambda1 = 0, lambda2 = NULL, lambda3 = NULL, lambda4 = NULL,
param = "fmkl", inverse.eps = 1e-08,
max.iterations = 500)
qgl(p, lambda1, lambda2 = NULL, lambda3 = NULL, lambda4 = NULL,
param = "fmkl")
rgl(n, lambda1=0, lambda2 = NULL, lambda3 = NULL, lambda4 = NULL,
param = "fmkl")x |
Vector of actual values for |
p |
Vector of probabilities for |
q |
Vector of quantiles for |
n |
Number of observations to be generated for |
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 |
lambda2 |
Scale parameter |
lambda3 |
First shape parameter |
lambda4 |
Second shape parameter |
param |
|
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 |
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.
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 observations. |
Freimer, M., Kollia, G., Mudholkar, G. S. & 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.
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)
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