GeneralisedLambdaDistribution {gld} | R Documentation |
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.
dgl(x, lambda1, lambda2, lambda3, lambda4, parameterisation="fmkl",inverse.eps=1e-8) qdgl(p, lambda1, lambda2, lambda3, lambda4, parameterisation="fmkl") qdgl.fmkl(p, lambda1, lambda2, lambda3, lambda4) qdgl.rs(p, lambda1, lambda2, lambda3, lambda4) pgl(q, lambda1, lambda2, lambda3, lambda4, parameterisation="fmkl",inverse.eps=1e-8) qgl(p, lambda1, lambda2, lambda3, lambda4, parameterisation="fmkl") qgl.fmkl(p, lambda1, lambda2, lambda3, lambda4) qgl.rs(p, lambda1, lambda2, lambda3, lambda4) rgl(n, lambda1, lambda2, lambda3, lambda4, parameterisation="fmkl")
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
parameterisation="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 the distribution and
quantile 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
,
is provided.
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.
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.
Robert King, robert.king@mailbox.gu.edu.au, http://www.ens.gu.edu.au/robertk/
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, 35473567.
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, 611642.
Ramberg, J. S. & Schmeiser, B. W. (1974), An approximate method for generating asymmetric random variables, Communications of the ACM 17, 7882.
http://www.ens.gu.edu.au/robertk/gld/
qgl(seq(0,1,0.02),0,1,0.123,-4.3)