GeneralizedLambda: The Generalized Lambda Distribution

In GeneralizedLambda: The Generalized Lambda Distribution

Description

Density, density quantile, distribution function, quantile function and random generation for the generalized lambda distribution (also known as the asymmetric lambda, or Tukey lambda). Provides for four different parameterisations, the FMKL (recommended), the RS, the VSK , the AS and a five parameter version of the FMKL, the FM5.

Usage

dgl(x, lambda1 = 0, lambda2 = 1, lambda3 = 1, lambda4 = 1,
    param = c(lambda1, lambda2, lambda3, lambda4),
    version = "FMKL", lambda5 = NULL,
    inverse.eps = .Machine$double.eps, max.iterations = 500)

dqgl(p, lambda1 = 0, lambda2 = 1, lambda3 = 1, lambda4 = 1,
     param = c(lambda1, lambda2, lambda3, lambda4),
     version = "FMKL", lambda5 = NULL)

pgl(q, lambda1 = 0, lambda2 = 1, lambda3 = 1, lambda4 = 1,
    param = c(lambda1, lambda2, lambda3, lambda4),
    version = "FMKL", lambda5 = NULL,
    inverse.eps = .Machine$double.eps, max.iterations = 500)

qgl(p, lambda1 = 0, lambda2 = 1, lambda3 = 1, lambda4 = 1,
    param = c(lambda1, lambda2, lambda3, lambda4),
    version = "FMKL", lambda5 = NULL, lower.tail = TRUE)

rgl(n, lambda1 = 0, lambda2 = 1, lambda3 = 1, lambda4 = 1,
    param = c(lambda1, lambda2, lambda3, lambda4),
    version = "FMKL", lambda5 = NULL)

Arguments

x,q	Vector of quantiles.
p	Vector of probabilities.
n	Number of observations.
param	vector of length 4 specifying the parameters of the generalized lambda distribution (see below for details). Note that in the VSK parameterisation, the four parameters are labelled alpha, beta, delta,lambda.
lambda1	lambda 1, the location parameter of the distribution alpha for the VSK parameterisation.
lambda2	lambda 2 - scale parameter beta for VSK
lambda3	lambda 3 - first shape parameter delta, a skewness parameter for VSK
lambda4	lambda 4 - second shape parameter lambda, a tail-shape parameter for VSK
lambda5	lambda 5 - a skewing parameter, in the fm5 parameterisation
version	Choose parameterisation (see below for details)
lower.tail	Logical. If TRUE, probabilities are P(X <= x), otherwise as P(X > x). FMKL uses Freimer, Mudholkar, Kollia and Lin (1988) (default). RS uses Ramberg and Schmeiser (1974). VSK uses VSK parameterisation, see van Staden and Loots (2009). AS uses asymmetry steepness fm5 uses the 5 parameter version of the FMKL parameterisation see Gilchrist (2000)
inverse.eps	Accuracy of calculation for the numerical determination of F(x), defaults to .Machine$double.eps
max.iterations	Maximum number of iterations in the numerical determination of F(x), defaults to 500

Details

The generalized 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 Q(u), the inverse of the distribution function.

The GeneralizedLambda 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:

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

for lambda2 > 0.

A second parameterisation, the RS, chosen by setting version="RS" is that due to Ramberg and Schmeiser (1974), with the quantile function:

Q(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 glCheckPars for details.

Another parameterisation, the VSK, chosen by setting version = "VSK" is due to van Staden and Loots (2009), with a quantile function:

Q(u) = alpha + beta ((1 - delta)(u^lambda - 1)/(lambda) - delta((1 - u)^lambda - 1)/lambda

for beta > 0 and -1 <= delta <= 1. (where the parameters appear in the par argument to the function in the order α,β,δ,λ). This parameterisation has simpler L-moments than other parameterisations and δ is a skewness parameter and λ is a tailweight parameter.

Another parameterisation, the FM5, chosen by setting version = "FM5" adds an additional skewness parameter to the FMKL parameterisation. This uses the same approach as that used by Gilchrist (2000) 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 numerical solutions to the quantile function, using the Newton-Raphson method.

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.

rgl generates random deviates.

Author(s)

Robert King, [e-mail], http://tolstoy.newcastle.edu.au/~rking/

Jeong Min Kim, [e-mail]

David Scott, [e-mail]

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) Modelling with Quantile Functions, Chapman \& Hall/CRC, London.

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

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.

Van Staden, Paul J., & M.T. Loots. (2009), Method of L-moment Estimation for the Generalized Lambda Distribution. In Proceedings of the Third Annual ASEARC Conference. Callaghan, NSW 2308 Australia: School of Mathematical and Physical Sciences, University of Newcastle.

Examples

qgl(seq(0, 1, 0.02), 0, 1, 0.123, -4.3,
    lower.tail = FALSE, version = "FMKL")
rgl(10, 0, 1.24, 1, 0.7, version = "VSK")
## Not run: 
  pgl(0.6, 1, 0.2, 0.3, -2, version = "RS") # Error

## End(Not run)

GeneralizedLambda documentation built on May 2, 2019, 4:50 p.m.