Package 'gld' reference manual

Title:	Estimation and Use of the Generalised (Tukey) Lambda Distribution
Description:	The generalised lambda distribution, or Tukey lambda distribution, provides a wide variety of shapes with one functional form. This package provides random numbers, quantiles, probabilities, densities and density quantiles for four different types of the distribution, the FKML (Freimer et al 1988), RS (Ramberg and Schmeiser 1974), GPD (van Staden and Loots 2009) and FM5 - see documentation for details. It provides the density function, distribution function, and Quantile-Quantile plots. It implements a variety of estimation methods for the distribution, including diagnostic plots. Estimation methods include the starship (all 4 types), method of L-Moments for the GPD and FKML types, and a number of methods for only the FKML type. These include maximum likelihood, maximum product of spacings, Titterington's method, Moments, Trimmed L-Moments and Distributional Least Absolutes.
Authors:	Robert King [aut, cre] , Benjamin Dean [aut], Sigbert Klinke [aut], Paul van Staden [aut]
Maintainer:	Robert King <[email protected]>
License:	GPL (>= 2)
Version:	2.6.6
Built:	2024-11-04 05:34:11 UTC
Source:	https://github.com/newystats/gld

Calculates Beta function for two identical parameters, allowing non-integer negative values

Description

By defining the Beta Function in terms of the Gamma Function,

$B(a,b)=\frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)}$

the function can be defined for non-integer negative values of a and b. The special case of this where $a=b$ is needed to calculate the standard errors of the L Moment estimates of the gpd type of the generalised lambda distribution, so this function carries out that calculation.

Usage

BetaLambdaLambda(lambda)
BetaLambdaLambda(lambda)

Arguments

lambda

A vector, each element of which is used for both arguments of the Beta function.

Details

NaN is returned for any negative integer elements of lambda.

Value

A vector the same length as lambda, containing Beta(lambda,lambda)

Author(s)

Robert King, [email protected], https://github.com/newystats/

Paul van Staden

References

https://github.com/newystats/gld/

Examples

BetaLambdaLambda(-0.3)
BetaLambdaLambda(-0.3)

Estimate parameters of the FKML parameterisation of the generalised lambda distribution

Description

Estimates parameters of the FKML parameterisation of the Generalised $\lambda$ Distribution. Five estimation methods are available; Numerical Maximum Likelihood, Maximum Product of Spacings, Titterington's Method, the Starship (also available in the starship function, which uses the same underlying code as this for the fkml parameterisation), Trimmed L-Moments, L-Moments, Distributional Least Absolutes, and Method of Moments.

Usage

fit.fkml(x, method = "ML", t1 = 0, t2 = 0, 
  l3.grid = c(-0.9, -0.5, -0.1, 0, 0.1, 0.2, 0.4, 0.8, 1, 1.5), 
  l4.grid = l3.grid, record.cpu.time = TRUE, optim.method = "Nelder-Mead",
  inverse.eps = .Machine$double.eps, optim.control=list(maxit=10000), 
  optim.penalty=1e20, return.data=FALSE)
fit.fkml(x, method = "ML", t1 = 0, t2 = 0, 
  l3.grid = c(-0.9, -0.5, -0.1, 0, 0.1, 0.2, 0.4, 0.8, 1, 1.5), 
  l4.grid = l3.grid, record.cpu.time = TRUE, optim.method = "Nelder-Mead",
  inverse.eps = .Machine$double.eps, optim.control=list(maxit=10000), 
  optim.penalty=1e20, return.data=FALSE)

Arguments

`x`	Data to be fitted, as a vector
`method`	A character string, to select the estimation method. One of: `ML` for numerical Maximum Likelihood, `MPS` or `MSP` for Maximum Spacings Product, `TM` for Titterington's Method, `SM` for Starship Method, `TL` for method of Trimmed L-moments, `Lmom` for method of L-moments, `DLA` for the method of Distributional Least Absolutes, or `Mom` for method of Moments.
`t1`	Number of observations to be trimmed from the left in the conceptual sample, $t_1$ (A non-negative integer, only used by TL-moment estimation, see details section)
`t2`	Number of observations to be trimmed from the right in the conceptual sample, $t_2$ (A non-negative integer, only used by TL-moment estimation, see details section). These two arguments are restricted by $t_1+t_2 < n$ , where n is the sample size
`l3.grid`	A vector of values to form the grid of values of $\lambda_3$ used to find a starting point for the optimisation.
`l4.grid`	A vector of values to form the grid of values of $\lambda_4$ used to find a starting point for the optimisation.
`record.cpu.time`	Boolean — should the CPU time used in fitting be recorded in the fitted model object?
`optim.method`	Optimisation method, use any of the options available under `method` of `optim`.
`inverse.eps`	Accuracy of calculation for the numerical determination of $F(x)$ , defaults to `.Machine$double.eps`.
`optim.control`	List of options for the optimisation step. See `optim` for details.
`optim.penalty`	The penalty to be added to the objective function if parameter values are proposed outside the allowed region
`return.data`	Logical: Should the function return the data (from the argument `data`)?

Details

Maximum Likelihood Estimation of the generalised lambda distribution (gld) proceeds by calculating the density of the data for candidate values of the parameters. Because the gld is defined by its quantile function, the method first numerically obtains F(x) by inverting Q(u), then obtains the density for that observation.

Maximum Product of Spacings estimation (sometimes referred to as Maximum Spacing Estimation, or Maximum Spacings Product) finds the parameter values that maximise the product of the spacings (the difference between successive depths, $F_\theta(x_{(i+1)})-F_\theta(x_{(i)})$ , where $F_\theta(x)$ is the distribution function for the candidate values of the parameters). See Dean (2013) and Cheng & Amin (1981) for details.

Titterington (1985) remarked that MPS effectively added an “extra observation”; there are N data points in the original sample, but N + 1 spacings in the expression maximised in MPS. Instead of using spacings between transformed data points, so method TM uses spacings between transformed, adjacently-averaged, data points. The spacings are given by $D_i=F_\theta(z_{(i)})-F_\theta(z_{(i-1)})$ , where $\alpha_1 = z_0 < z_1 < \dots < z_n = \alpha_2$ and $z_i = (x_{(i)} +x_{(i+1)})/2$ for i = 1,2, ... n-1 (where $\alpha_1$ and $\alpha_2$ are the lower and upper bounds on the support of the distribution). This reduces the number of spacings to $n$ and achieves concordance with the original sample size. See Titterington (1985) and Dean (2013) for details.

The starship is built on the fact that the $g\lambda d$ is a transformation of the uniform distribution. Thus the inverse of this transformation is the distribution function for the gld. The starship method applies different values of the parameters of the distribution to the distribution function, calculates the depths q corresponding to the data and chooses the parameters that make these calculated depths closest (as measured by the Anderson-Darling statistic) to a uniform distribution. See King & MacGillivray (1999) for details.

TL-Moment estimation chooses the values of the parameters that minimise the difference between the sample Trimmed L-Moments of the data and the Trimmed L-Moments of the fitted distribution. TL-Moments are based on inflating the conceptual sample size used in the definition of L-Moments. The t1 and t2 arguments to the function define the extent of trimming of the conceptual sample. Thus, the default values of t1=0 and t2=0 reduce the TL-Moment method to L-Moment estimation. t1 and t2 give the number of observations to be trimmed (from the left and right respectively) from the conceptual sample of size $n+t_1+t_2$ . These two arguments should be non-negative integers, and $t_1+t_2 < n$ , where n is the sample size. See Elamir and Seheult (2003) for more on TL-Moments in general, Asquith, (2007) for TL-Moments of the RS parameterisation of the gld and Dean (2013) for more details on TL-Moment estimation of the gld.

The method of distributional least absolutes (DLA) minimises the sum of absolute deviations between the order statistics and their medians (based on the candidate parameters). See Dean (2013) for more information.

Moment estimation chooses the values of the parameters that minimise the (sum of the squared) difference between the first four sample moments of the data and the first four moments of the fitted distribution.

Value

fit.fkml returns an object of class "starship" (regardless of the estimation method used).

print prints the estimated values of the parameters, while summary.starship prints these by default, but can also provide details of the estimation process (from the components grid.results, data and optim detailed below).

The value of fit.fkml is a list containing the following components:

`lambda`	A vector of length 4, giving the estimated parameters, in order, $\lambda_1$ - location parameter $\lambda_2$ - scale parameter $\lambda_3$ - first shape parameter $\lambda_4$ - second shape parameter
`grid.results`	output from the grid search
`optim`	output from the optim search, `optim` for details
`cpu`	A vector showing the computing time used, returned if `record.cpu.time` is TRUE
`data`	The data, if `return.data` is TRUE

Author(s)

Robert King, [email protected], https://github.com/newystats/

Ben Dean, University of Newcastle [email protected]

References

Asquith, W. H. (2007), L-Moments and TL-Moments of the Generalized Lambda Distribution, Computational Statistics & Data Analysis, 51, 4484–4496.

Cheng, R.C.H. & Amin, N.A.K. (1981), Maximum Likelihood Estimation of Parameters in the Inverse Gaussian Distribution, with Unknown Origin, Technometrics, 23(3), 257–263. https://www.jstor.org/stable/1267789

Dean, B. (2013) Improved Estimation and Regression Techniques with the Generalised Lambda Distribution, PhD Thesis, University of Newcastle https://nova.newcastle.edu.au/vital/access/manager/Repository/uon:13503

Elamir, E. A. H., and Seheult, A. H. (2003), Trimmed L-Moments, Computational Statistics & Data Analysis, 43, 299–314.

King, R.A.R. & MacGillivray, H. L. (1999), A starship method for fitting the generalised $\lambda$ distributions, Australian and New Zealand Journal of Statistics 41, 353–374.

Titterington, D. M. (1985), Comment on ‘Estimating Parameters in Continuous Univariate Distributions’, Journal of the Royal Statistical Society, Series B, 47, 115–116.

Examples

example.data <- rgl(200,c(3,1,.4,-0.1),param="fkml")
example.fit <- fit.fkml(example.data,"MSP",return.data=TRUE)
print(example.fit)
summary(example.fit)
plot(example.fit,one.page=FALSE)
example.data <- rgl(200,c(3,1,.4,-0.1),param="fkml")
example.fit <- fit.fkml(example.data,"MSP",return.data=TRUE)
print(example.fit)
summary(example.fit)
plot(example.fit,one.page=FALSE)

Method of moments estimation for the FKML type of the generalised lambda distribution

Description

Estimates parameters of the generalised lambda distribution (FKML type) using the Method of Moments, on the basis of moments calculated from data, or moment values (mean, variance, skewness ratio and kurtosis ratio (note, not the excess kurtosis)).

Usage

fit.fkml.moments(data,na.rm=TRUE,
  optim.method="Nelder-Mead",
  optim.control= list(), starting.point = c(0,0))
fit.fkml.moments.val(moments=c(mean=0, variance=1, skewness=0,
  kurtosis=3), optim.method="Nelder-Mead", optim.control= list(), 
  starting.point = c(0,0))
fit.fkml.moments(data,na.rm=TRUE,
  optim.method="Nelder-Mead",
  optim.control= list(), starting.point = c(0,0))
fit.fkml.moments.val(moments=c(mean=0, variance=1, skewness=0,
  kurtosis=3), optim.method="Nelder-Mead", optim.control= list(), 
  starting.point = c(0,0))

Arguments

`data`	A vector of data
`na.rm`	Logical - should NAs be removed from the data - if FALSE, any NAs in the data will cause an error
`moments`	A vector of length 4, consisting of the mean, variance and moment ratios for skewness and kurtosis (do not subtract 3 from the kurtosis ratio)
`optim.method`	Optimisation method for `optim` to use, defaults to Nelder-Mead
`optim.control`	argument `control`, passed to `optim`.
`starting.point`	a vector of length 2, giving the starting value for $\lambda_3$ and $\lambda_4$ .

Details

Estimates parameters of the generalised lambda distribution (FKML type) using Method of Moments on the basis of moment values (mean, variance, skewness ratio and kurtosis ratio). Note this is the fourth central moment divided by the second central moment, without subtracting 3. fit.fkml.moments will estimates using the method of moments for a dataset, including calculating the sample moments. This function uses optim to find the parameters that minimise the sum of squared differences between the skewness and kurtosis sample ratios and their counterpart expressions for those ratios on the basis of the parameters $\lambda_3$ and $\lambda_4$ . On the basis of these estimates (and the mean and variance), this function then estimates $\hat\lambda_2$ and then $\hat\lambda_1$ .

Note that the first 4 moments don't uniquely identify members of the generalised $\lambda$ distribution. Typically, for a set of moments that correspond to a unimodal gld, there is another set of parameters that give a distrbution with the same first 4 moments. This other distribution has a truncated appearance (that is, the distribution has finite support and the density is non-zero at the end points). See the examples below.

Value

A vector containing the parameters of the FKML type generalised lambda; $\lambda_1$ - location parameter $\lambda_2$ - scale parameter $\lambda_3$ - first shape parameter $\lambda_4$ - second shape parameter (See gld for more details)

Author(s)

Robert King, [email protected], https://github.com/newystats/

Sigbert Klinke

Paul van Staden

References

Au-Yeung, Susanna W. M. (2003) Finding Probability Distributions From Moments, Masters thesis, Imperial College of Science, Technology and Medicine (University of London), Department of Computing

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.

Lakhany, Asif and Mausser, Helmut (2000) Estimating the parameters of the generalized lambda distribution, Algo Research Quarterly, 3(3):47–58

van Staden, Paul (2013) Modeling of generalized families of probability distributions inthe quantile statistical universe, PhD thesis, University of Pretoria. https://repository.up.ac.za/handle/2263/40265

https://github.com/newystats/gld/

Examples

# Moment estimate
example.data = rgl(n=400,lambda1=c(0,1,0.4,0),
  param="fkml")
fit.fkml.moments(example.data)
# Approximation to the standard normal distribution
norm.approx <- fit.fkml.moments.val(c(0,1,0,3))
norm.approx
# Another distribution with the same moments
another <- fit.fkml.moments.val(c(0,1,0,3),start=c(2,2))
another
# Compared
plotgld(norm.approx$lambda,ylim=c(0,0.75),main="Approximation to the standard normal",
  sub="and another GLD with the same first 4 moments")
plotgld(another$lambda,add=TRUE,col=2)
# Moment estimate
example.data = rgl(n=400,lambda1=c(0,1,0.4,0),
  param="fkml")
fit.fkml.moments(example.data)
# Approximation to the standard normal distribution
norm.approx <- fit.fkml.moments.val(c(0,1,0,3))
norm.approx
# Another distribution with the same moments
another <- fit.fkml.moments.val(c(0,1,0,3),start=c(2,2))
another
# Compared
plotgld(norm.approx$lambda,ylim=c(0,0.75),main="Approximation to the standard normal",
  sub="and another GLD with the same first 4 moments")
plotgld(another$lambda,add=TRUE,col=2)

Estimate parameters of the GPD type generalised lambda distribution

Description

Estimates parameters of the GPD type generalised $\lambda$ Distribution. Estimation is via method of L moments or the starship method.

The Method of L-Moments estimates for the GPD type are the only estimates for any generalised lambda distribution type with closed form expressions, and the only with algebraic results for standard errors of the estimates.

For further details on the starship method, see starship.

Usage

fit.gpd(x, method = "LM", na.rm = TRUE, record.cpu.time = TRUE, return.data = FALSE,
    LambdaZeroEpsilon=1e-15)
fit.gpd.lmom(data, na.rm = TRUE,LambdaZeroEpsilon=1e-15)
fit.gpd.lmom.given(lmoms, n = NULL,LambdaZeroEpsilon=1e-15)
fit.gpd(x, method = "LM", na.rm = TRUE, record.cpu.time = TRUE, return.data = FALSE,
    LambdaZeroEpsilon=1e-15)
fit.gpd.lmom(data, na.rm = TRUE,LambdaZeroEpsilon=1e-15)
fit.gpd.lmom.given(lmoms, n = NULL,LambdaZeroEpsilon=1e-15)

Arguments

`x`	Data to be fitted, as a vector
`method`	A character string, to select the estimation method. The available methods are 1. "LM" for Method of L-Moments, 2. "SM" or "starship" for the Starship method.
`na.rm`	Logical: Should missing values be removed?
`record.cpu.time`	Logical: should the CPU time used in fitting be recorded in the fitted model object?
`return.data`	Logical: Should the function return the data (from the argument `x`)?
`data`	Data to be fitted, as a vector
`lmoms`	A numeric vector containing two L-moments and two L-moment ratios, in the order $l_1,l_2,t_3,t_4$ .
`n`	the sample size, defaults to NULL
`LambdaZeroEpsilon`	tolerance for lambda estimate of zero

Details

The starship method calls the starship function - see its help for more details.

The method of L-Moments equates sample L-Moments with expressions for the L-Moments of the GPD type GLD. Closed form expressions exist to give these estimates.

For many values there are two possible estimates for the same L Moment values, one in each of two regions of the GPD GLD parameter space, denoted region A and region B in van Staden (2013). More details on these regions can be found on page 154 of van Staden (2013).

If the 4th L-Moment ratio, $\tau_4$ is less than the minimum value that $\tau_4$ can obtain for the GPD generalised lambda distribution;

$\tau_4^{(min)}=\frac{12-5\sqrt{6}}{12+5\sqrt{6}} \approx -0.0102051,$

there is no possible L-Moment estimate (from either region A or B), and this function returns NA for the estimates.

When estimating from data, or for given L-Moments with $n$ given, standard errors of the estimates are calculated if possible (standard errors are only finite if $\lambda > -0.5$ ).

If $\lambda$ is zero, the GPD gld is a special case the Quantile Based Skew Logistic Distribution. If the estimated $\lambda$ is within LambdaZeroEpsilon of zero, standard errors for alpha, beta and delta are calculated for the Quantile Based Skew Logistic Distribution and NA is returned as the standard error of $\lambda$ .

Value

These functions return an object of class "GldGPDFit". It is a list, containing these components (optional components noted here);

`estA`	The estimate in region A. This will be NULL if there is no estimate in region A
`estB`	The estimate in region B. This will be NULL if there is no estimate in region B
`warn`	(only if estA and estB are both NULL), the reason there are no estimates. If this is the case, the function also issues a `warning`.
`cpu`	A vector showing the computing time used, returned if `record.cpu.time` is TRUE (only for `fit.gpd`).
`data`	The data, if `return.data` is TRUE (only for `fit.gpd`).
`param`	The character `"gpd"`, indicating the GPD type of the generalised lambda distribution.
`starship`	The value returned by `starship`, if the `starship` method is used.

Each of the estimate elements (if they are not NULL) are either a vector of length 4, or a 4 by 2 matrix if standard errors are calculated. The elements of the vector, or rows of the matrix are the estimated parameters, in order;

`alpha`	location parameter
`beta`	scale parameter
`delta`	skewness parameter
`lambda`	kurtosis parameter

The columns of the matrix are the parameter, and its standard error.

Author(s)

Robert King, [email protected], https://github.com/newystats/

Paul van Staden

References

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

fit.gpd.lmom.given(c(1,.3,.6,.8))
example.data = rgl(n=300,c(5,2,0.8,-0.1),param="gpd")
fit.gpd(example.data)
fit.gpd(example.data,method="starship")
fit.gpd.lmom.given(c(1,.3,.6,.8))
example.data = rgl(n=300,c(5,2,0.8,-0.1),param="gpd")
fit.gpd(example.data)
fit.gpd(example.data,method="starship")

The Generalised Lambda Distribution

Description

Density, density quantile, distribution function, quantile function and random generation for the generalised lambda distribution (also known as the asymmetric lambda, or Tukey lambda). Provides for four different parameterisations, the fmkl (recommended), the rs, the gpd and a five parameter version of the FMKL, the fm5.

Usage

dgl(x, lambda1 = 0, lambda2 = NULL, lambda3 = NULL, lambda4 = NULL, 
  param = "fkml", lambda5 = NULL, inverse.eps = .Machine$double.eps,
  max.iterations = 500)
dqgl(p, lambda1, lambda2 = NULL, lambda3 = NULL, lambda4 = NULL, 
  param = "fkml", lambda5 = NULL)
pgl(q, lambda1 = 0, lambda2 = NULL, lambda3 = NULL, lambda4 = NULL, 
  param = "fkml", lambda5 = NULL, inverse.eps = .Machine$double.eps,
  max.iterations = 500)
qgl(p, lambda1, lambda2 = NULL, lambda3 = NULL, lambda4 = NULL,
  param = "fkml", lambda5 = NULL)
rgl(n, lambda1=0, lambda2 = NULL, lambda3 = NULL, lambda4 = NULL,
  param = "fkml", lambda5 = NULL)
dgl(x, lambda1 = 0, lambda2 = NULL, lambda3 = NULL, lambda4 = NULL, 
  param = "fkml", lambda5 = NULL, inverse.eps = .Machine$double.eps,
  max.iterations = 500)
dqgl(p, lambda1, lambda2 = NULL, lambda3 = NULL, lambda4 = NULL, 
  param = "fkml", lambda5 = NULL)
pgl(q, lambda1 = 0, lambda2 = NULL, lambda3 = NULL, lambda4 = NULL, 
  param = "fkml", lambda5 = NULL, inverse.eps = .Machine$double.eps,
  max.iterations = 500)
qgl(p, lambda1, lambda2 = NULL, lambda3 = NULL, lambda4 = NULL,
  param = "fkml", lambda5 = NULL)
rgl(n, lambda1=0, lambda2 = NULL, lambda3 = NULL, lambda4 = NULL,
  param = "fkml", 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`, `rs` and `gpd` 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 ( $\alpha$ for the `gpd` parameterisation). 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. Note also that in the `gpd` parameterisation, the four parameters are labelled $\alpha, \beta, \delta, \lambda$ .
`lambda2`	$\lambda_2$ - scale parameter ( $\beta$ for `gpd`)
`lambda3`	$\lambda_3$ - first shape parameter ( $\delta$ , a skewness parameter for `gpd`)
`lambda4`	$\lambda_4$ - second shape parameter ( $\lambda$ , a tail-shape parameter for `gpd`)
`lambda5`	$\lambda_5$ - a skewing parameter, in the fm5 parameterisation
`param`	choose parameterisation (see below for details) `fmkl` uses Freimer, Mudholkar, Kollia and Lin (1988) (default). `rs` uses Ramberg and Schmeiser (1974) `gpd` uses GPD parameterisation, see van Staden and Loots (2009) `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 `.Machine$double.eps`. You may wish to make this a larger number to speed things up for large samples.
`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 (Q(u)), 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:

$Q(u)= \lambda_1 + { { \frac{u^{\lambda_3}-1}{\lambda_3} - % \frac{(1-u)^{\lambda_4}-1}{\lambda_4} } \over \lambda_2 } %$

for $\lambda_2 > 0$ .

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

$Q(u)= \lambda_1 + \frac{u^{\lambda_3} - (1-u)^{\lambda_4}} % {\lambda_2}$

This parameterisation has a complex series of rules determining which values of the parameters produce valid statistical distributions. See gl.check.lambda for details.

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

$Q(u) = \alpha + \beta ((1-\delta)\frac{(u^\lambda -1)}{\lambda} - \delta\frac{((1-u)^\lambda -1)}{\lambda} )$

for $\beta > 0$ and $-1 \leq \delta \leq 1$ . (where the parameters appear in the par argument to the function in the order $\alpha,\beta,\delta,\lambda$ ). This parameterisation has simpler L-moments than other parameterisations and $\delta$ is a skewness parameter and $\lambda$ is a tailweight parameter.

Another 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 Gilchrist (2000) for the RS parameterisation. The quantile function is

$F^{-1}(u)= \lambda_1 + { { \frac{(1-\lambda_5)(u^{\lambda_3}-1)}% {\lambda_3} - \frac{(1+\lambda_5)((1-u)^{\lambda_4}-1)}{\lambda_4} } % \over \lambda_2 }$

for $\lambda_2 > 0$ and $-1 \leq \lambda_5 \leq 1$ .

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. Since the density quantile function, $f(F^{-1}(u))$ , does exist, an additional function, dqgl, 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 $F^{-1}(u)=x$ ),

qdgl gives the quantile density,

pgl gives the distribution function (based on a numerical solution to $F^{-1}(u)=x$ ),

qgl gives the quantile function, and

rgl generates random deviates.

Author(s)

Robert King, [email protected], https://github.com/newystats/

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 and 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 fi ts, 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.

https://github.com/newystats/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")
qgl(seq(0,1,0.02),0,1,0.123,-4.3)
pgl(seq(-2,2,0.2),0,1,-.1,-.2,param="fmkl")

Function to check the validity of parameters of the generalized lambda distribution

Description

Checks the validity of parameters of the generalized lambda. The tests are simple for the FMKL, FM5 and GPD types, and much more complex for the RS parameterisation.

Usage

gl.check.lambda(lambdas, lambda2 = NULL, lambda3 = NULL, lambda4 = NULL, param = "fkml", 
  lambda5 = NULL, vect = FALSE)
gl.check.lambda(lambdas, lambda2 = NULL, lambda3 = NULL, lambda4 = NULL, param = "fkml", 
  lambda5 = NULL, vect = FALSE)

Arguments

`lambdas`	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 ( $\beta$ for `gpd`)
`lambda3`	$\lambda_3$ - first shape parameter ( $\delta$ , skewness parameter for `gpd`)
`lambda4`	$\lambda_4$ - second shape parameter ( $\lambda$ , kurtosis parameter for `gpd`)
`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)
`vect`	A logical, set this to TRUE if the parameters are given in the vector form (it turns off checking of the format of `lambdas` and the other lambda arguments

Details

See GeneralisedLambdaDistribution for details on the generalised lambda distribution. This function determines the validity of parameters of the distribution.

The FMKL parameterisation gives a valid statistical distribution for any real values of $\lambda_1$ , $\lambda_3$ , $\lambda_4$ and any positive real values of $\lambda_2$ .

The FM5 parameterisation gives statistical distribution for any real values of $\lambda_1$ , $\lambda_3$ , $\lambda_4$ , any positive real values of $\lambda_2$ and values of $\lambda_5$ that satisfy $-1 \leq \lambda_5 \leq 1$ .

For the RS parameterisation, the combinations of parameters value that give valid distributions are the following (the region numbers in the table correspond to the labelling of the regions in Ramberg and Schmeiser (1974) and Karian, Dudewicz and McDonald (1996)):

region	$\lambda_1$	$\lambda_2$	$\lambda_3$	$\lambda_4$	note
1	all	$<0$	$< -1$	$> 1$
2	all	$<0$	$> 1$	$< -1$
3	all	$>0$	$\ge 0$	$\ge 0$	one of $\lambda_3$ and $\lambda_4$ must be non-zero
4	all	$<0$	$\le 0$	$\le 0$	one of $\lambda_3$ and $\lambda_4$ must be non-zero
5	all	$<0$	$> -1$ and $< 0$	$>1$	equation 1 below must also be satisfied
6	all	$<0$	$>1$	$> -1$ and $< 0$	equation 2 below must also be satisfied

Equation 1

$\frac{(1-\lambda_3) ^{1-\lambda_3}(\lambda_4-1)^{\lambda_4-1}} {(\lambda_4-\lambda_3)^{\lambda_4-\lambda_3}} < - \frac{\lambda_3}{\lambda_4}$

Equation 2

$\frac{(1-\lambda_4) ^{1-\lambda_4}(\lambda_3-1)^{\lambda_3-1}} {(\lambda_3-\lambda_4)^{\lambda_3-\lambda_4}} < - \frac{\lambda_4}{\lambda_3}$

The GPD type gives a valid distribution provided $\beta$ is positive and $0 \leq \delta \leq 1$ .

Value

This logical function takes on a value of TRUE if the parameter values given produce a valid statistical distribution and FALSE if they don't

Note

The complex nature of the rules in this function for the RS parameterisation are the reason for the invention of the FMKL parameterisation and its status as the default parameterisation in the other generalized lambda functions.

Author(s)

Robert King, [email protected], https://github.com/newystats/

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.

https://github.com/newystats/gld/

Examples

gl.check.lambda(c(0,1,.23,4.5),vect=TRUE) ## TRUE
gl.check.lambda(c(0,-1,.23,4.5),vect=TRUE) ## FALSE 
gl.check.lambda(c(0,1,0.5,-0.5),param="rs",vect=TRUE) ## FALSE
gl.check.lambda(c(0,2,1,3.4,1.2),param="fm5",vect=TRUE) ## FALSE
gl.check.lambda(c(0,1,.23,4.5),vect=TRUE) ## TRUE
gl.check.lambda(c(0,-1,.23,4.5),vect=TRUE) ## FALSE 
gl.check.lambda(c(0,1,0.5,-0.5),param="rs",vect=TRUE) ## FALSE
gl.check.lambda(c(0,2,1,3.4,1.2),param="fm5",vect=TRUE) ## FALSE

Deprecated functions

Description

qdgl: This calculates the density quantile function of the GLD, so it has been renamed dqgl.

Calculate L-Moments of the GPD type generalised lambda distribution for given parameter values

Description

Calculates the first four L-Moments of the GPD type generalised $\lambda$ distribution for given parameter values.

Usage

  gld.lmoments(pars,order=1:4,ratios=TRUE,type="GPD",param=NULL)
gld.lmoments(pars,order=1:4,ratios=TRUE,type="GPD",param=NULL)

Arguments

`pars`	A vector of length 4, giving the parameters of the GPD type generalised lambda distribution, consisting of; $\alpha$ location parameter $\beta>0$ scale parameter $0 \leq \delta \leq 1$ skewness parameter $\lambda$ kurtosis parameter
`order`	Integers to select the orders of L-moments to calculate. Currently this function only calculates for orders 1 to 4.
`type`	choose the type of generalised lambda distribution. Currently `gld.lmoments` only supports `GPD` which uses van Staden and Loots (2009) (default).
`ratios`	Logical. TRUE gives L-moment ratios for skewness and kurtosis ( $\tau_3$ and $\tau_4$ ) (and all higher orders), FALSE gives the requested L-moments instead.
`param`	alias for the type argument. The `type` argument is preferred.

Details

The GPD type generalised $\lambda$ distribution was introduced by van Staden and Loots (2009). It has explicit parameters for skewness and kurtosis, and closed form estimates for L-moment estimates of the parameters.

In the limit, as the kurtosis parameter, $\lambda$ , goes to zero, the distribution approaches the skew logistic distribution of van Staden and King (2013). See the sld package for this distribution.

Value

A vector containing the selected L-moments of the GPD type generalised lambda. If ratio is true, the vector contains L-Moment ratios for orders 3 and over, otherwise all values are L-Moments.

Author(s)

Robert King, [email protected], https://github.com/newystats/

Paul van Staden

References

van Staden, P.J. and King, Robert A.R. (2015) The quantile-based skew logistic distribution, Statistics and Probability Letters 96 109–116. doi:10.1016/j.spl.2014.09.001

van Staden, Paul J. 2013 Modeling of generalized families of probability distribution in the quantile statistical universe. PhD thesis, University of Pretoria. https://repository.up.ac.za/handle/2263/40265

Quantile based Skew logistic distribution

Generalised Lambda Distribution

https://github.com/newystats/gld/

Examples

gld.lmoments(c(0,1,0.5,0.23))
gld.lmoments(c(0,1,0,0.23))
gld.lmoments(c(0,1,0.5,0.7))
gld.lmoments(c(0,1,0.5,0.23))
gld.lmoments(c(0,1,0,0.23))
gld.lmoments(c(0,1,0.5,0.7))

Calculate moments of the FKML type of the generalised lambda distribution for given parameter values

Description

Calculates the mean, variance, skewness ratio and kurtosis ratio of the generalised $\lambda$ distribution for given parameter values.

Usage

  gld.moments(par,type="fkml",ratios=TRUE)
gld.moments(par,type="fkml",ratios=TRUE)

Arguments

`par`	A vector of length 4, giving the parameters of the generalised lambda distribution, consisting of; $\lambda_1$ location parameter $\lambda_2$ - scale parameter $\lambda_3$ - first shape parameter $\lambda_4$ - second shape parameter
`type`	choose the type of generalised lambda distribution. Currently `gld.moments` only supports `fkml` which uses Freimer, Kollia, Mudholkar, and Lin (1988) (default).
`ratios`	Logical. TRUE to give moment ratios for skewness and kurtosis, FALSE to give the third and fourth central moments instead.

Details

The FKML type of the generalised $\lambda$ distribution was introduced by Freimer et al (1988) who gave expressions for the moments. In the limit, as the shape parameters ( $\lambda_3$ and $\lambda_4$ ) go to zero, the distribution is defined using limit results. The moments in these limiting cases were given by van Staden (2013). This function calculates the first 4 moments.

See pages 96–97 of van Staden (2013) for the full expressions for these moments.

Value

A vector containing the first four moments of the FKML type generalized lambda. If ratio is true, the vector contains the mean, variance, skewness ratio and kurtosis ratio. If ratio is false, the vector contains the mean, variance, third central moment and fourth central moment.

Author(s)

Robert King, [email protected], https://github.com/newystats/

Sigbert Klinke

Paul van Staden

References

Au-Yeung, Susanna W. M. (2003) Finding Probability Distributions From Moments, Masters thesis, Imperial College of Science, Technology and Medicine (University of London), Department of Computing

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.

Lakhany, Asif and Mausser, Helmut (2000) Estimating the parameters of the generalized lambda distribution, Algo Research Quarterly, 3(3):47–58

van Staden, Paul J. (2013) Modeling of generalized families of probability distributions inthe quantile statistical universe, PhD thesis, University of Pretoria. https://repository.up.ac.za/handle/2263/40265

https://github.com/newystats/gld/

Examples

gld.moments(c(0,1.463551,0.1349124,0.1349124))
gld.moments(c(0,1.813799,0,0))
gld.moments(c(0,1,0,3))
gld.moments(c(0,1.463551,0.1349124,0.1349124))
gld.moments(c(0,1.813799,0,0))
gld.moments(c(0,1,0,3))

Plots to compare a fitted generalised lambda distribution to data

Description

Plots to compare a fitted Generalised Lambda Distribution to data. This works with fitted gld objects from starship or fit.fkml.

Usage

## S3 method for class 'starship'
plot(x, data, ask = NULL, one.page = FALSE, 
    breaks = "Sturges", plot.title = "default",...)
## S3 method for class 'starship'
plot(x, data, ask = NULL, one.page = FALSE, 
    breaks = "Sturges", plot.title = "default",...)

Arguments

`x`	An object of class `starship`. These are produced by the fitting functions `fit.fkml` and `starship`.
`data`	Data to which the gld was fitted. Leave this as NULL if the return.data argument was TRUE in the call that created (`starship` or `fit.fkml`)
`ask`	Ask for user input before next plot. The default of NULL changes to TRUE if one.page is FALSE and `plot` is called interactively, otherwise it changes to FALSE this is then passed to `par(ask)`. Does not permanently change this setting. The argument is ignored if `one.page` is TRUE
`one.page`	If TRUE, put the two plots on one page using `par(mfrow=c(2,1))`. Does not permanently change this setting.
`breaks`	Control the number of histogram bins — passed to `hist`.
`plot.title`	Main title for histogram and QQ — passed to `hist(main=)` and `qqgl(main=)`. If you set this to `"default"`, it will include the fitting method and gld type, for example “Starship fit of FMKL type GLD”.
`...`	arguments passed to `plot` AND `hist`

Details

This function produces a Q-Q plot of the data against the fitted generalised lambda distribution and a histogram of the data overlaid with the fitted probability density function.

Author(s)

Robert King, [email protected], https://github.com/newystats/

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.

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

King, R.A.R. & MacGillivray, H. L. (1999), A starship method for fitting the generalised $\lambda$ distributions, Australian and New Zealand Journal of Statistics 41, 353–374

https://github.com/newystats/gld/

Examples

data <- rgl(100,0,1,.2,.2)
starship.result <- starship(data,optim.method="Nelder-Mead",initgrid=list(lcvect=(0:4)/10,
ldvect=(0:4)/10),return.data=TRUE)
plot(starship.result)
data <- rgl(100,0,1,.2,.2)
starship.result <- starship(data,optim.method="Nelder-Mead",initgrid=list(lcvect=(0:4)/10,
ldvect=(0:4)/10),return.data=TRUE)
plot(starship.result)

Plots of density and distribution function for the generalised lambda distribution

Description

Produces plots of density and distribution function for the generalised lambda distribution. Although you could use plot(function(x)dgl(x)) to do this, the fact that the density and quantiles of the generalised lambda are defined in terms of the depth, $u$ , means that a seperate function that uses the depths to produce the values to plot is more efficient

Usage

plotgld(lambda1 = 0, lambda2 = NULL, lambda3 = NULL, lambda4 = NULL, 
  param = "fmkl", lambda5 = NULL, add = NULL, truncate = 0, 
  bnw = FALSE, col.or.type = 1, granularity = 10000, xlab = "x", 
  ylab = NULL, quant.probs = seq(0,1,.25), new.plot = NULL,  ...)
plotglc(lambda1 = 0, lambda2 = NULL, lambda3 = NULL, lambda4 = NULL, 
  param = "fmkl", lambda5 = NULL, granularity = 10000, xlab = "x", 
  ylab = "cumulative probability", add = FALSE, ...)
plotgld(lambda1 = 0, lambda2 = NULL, lambda3 = NULL, lambda4 = NULL, 
  param = "fmkl", lambda5 = NULL, add = NULL, truncate = 0, 
  bnw = FALSE, col.or.type = 1, granularity = 10000, xlab = "x", 
  ylab = NULL, quant.probs = seq(0,1,.25), new.plot = NULL,  ...)
plotglc(lambda1 = 0, lambda2 = NULL, lambda3 = NULL, lambda4 = NULL, 
  param = "fmkl", lambda5 = NULL, granularity = 10000, xlab = "x", 
  ylab = "cumulative probability", add = FALSE, ...)

Arguments

`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)
`add`	a logical value describing whether this should add to an existing plot (using `lines`) or produce a new plot (using `plot`). Defaults to FALSE (new plot) if both `add` and `new.plot` are NULL.
`truncate`	for `plotgld`, a minimum density value at which the plot should be truncated.
`bnw`	a logical value, true for a black and white plot, with different densities identified using line type (`lty`), false for a colour plot, with different densities identified using line colour (`col`)
`col.or.type`	Colour or type of line to use
`granularity`	Number of points to calculate quantiles and density at — see details
`xlab`	X axis label
`ylab`	Y axis label
`quant.probs`	Quantiles of distribution to return (see value below). Set to NULL to suppress this return entirely.
`new.plot`	a logical value describing whether this should produce a new plot (using `plot`), or add to an existing plot (using `lines`). Ignored if `add` is set.
`...`	arguments that get passed to `plot` if this is a new plot

Details

The generalised lambda distribution is defined in terms of its quantile function. The density of the distribution is available explicitly as a function of depths, $u$ , but not explicitly available as a function of $x$ . This function calculates quantiles and depths as a function of depths to produce a density plot plotgld or cumulative probability plot plotglc.

The plot can be truncated, either by restricting the values using xlim — see par for details, or by the truncate argument, which specifies a minimum density. This is recommended for graphs of densities where the tail is very long.

Value

A number of quantiles from the distribution, the default being the minimum, maximum and quartiles.

Author(s)

Robert King, [email protected], https://github.com/newystats/

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.

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

Karian, Z.E. & Dudewicz, E.J. (2000), Fitting Statistical Distributions to Data: The generalised Lambda Distribution and the Generalised Bootstrap Methods, CRC Press.

https://github.com/newystats/gld/

Examples

plotgld(0,1.4640474,.1349,.1349,main="Approximation to Standard Normal", 
sub="But you can see this isn't on infinite support")

plotgld(1.42857143,1,.7,.3,main="The whale")
plotglc(1.42857143,1,.7,.3)
plotgld(0,-1,5,-0.3,param="rs")
plotgld(0,-1,5,-0.3,param="rs",xlim=c(1,2))
# A bizarre shape from the RS paramterisation
plotgld(0,1,5,-0.3,param="fmkl")
plotgld(10/3,1,.3,-1,truncate=1e-3)

plotgld(0,1,.0742,.0742,col.or.type=2,param="rs",
main="All distributions have the same moments",
sub="The full Range of all distributions is shown")
plotgld(0,1,6.026,6.026,col.or.type=3,new.plot=FALSE,param="rs")
plotgld(0,1,35.498,2.297,col.or.type=4,new.plot=FALSE,param="rs")
legend(0.25,3.5,lty=1,col=c(2,3,4),legend=c("(0,1,.0742,.0742)",
"(0,1,6.026,6.026)","(0,1,35.498,2.297)"),cex=0.9)
# An illustration of problems with moments as a method of characterising shape

plotgld(0,1.4640474,.1349,.1349,main="Approximation to Standard Normal", 
sub="But you can see this isn't on infinite support")

plotgld(1.42857143,1,.7,.3,main="The whale")
plotglc(1.42857143,1,.7,.3)
plotgld(0,-1,5,-0.3,param="rs")
plotgld(0,-1,5,-0.3,param="rs",xlim=c(1,2))
# A bizarre shape from the RS paramterisation
plotgld(0,1,5,-0.3,param="fmkl")
plotgld(10/3,1,.3,-1,truncate=1e-3)

plotgld(0,1,.0742,.0742,col.or.type=2,param="rs",
main="All distributions have the same moments",
sub="The full Range of all distributions is shown")
plotgld(0,1,6.026,6.026,col.or.type=3,new.plot=FALSE,param="rs")
plotgld(0,1,35.498,2.297,col.or.type=4,new.plot=FALSE,param="rs")
legend(0.25,3.5,lty=1,col=c(2,3,4),legend=c("(0,1,.0742,.0742)",
"(0,1,6.026,6.026)","(0,1,35.498,2.297)"),cex=0.9)
# An illustration of problems with moments as a method of characterising shape

Print (or summarise) the results of a starship estimation

Description

Print (or summarise) the results of a starship estimation of the parameters of the Generalised Lambda Distribution

Usage

## S3 method for class 'starship'
summary(object, ...)

## S3 method for class 'starship'
print(x, digits = max(3, getOption("digits") - 3), ...)
## S3 method for class 'starship'
summary(object, ...)

## S3 method for class 'starship'
print(x, digits = max(3, getOption("digits") - 3), ...)

Arguments

`x`	An object of class `starship`.
`object`	An object of class `starship`.
`digits`	minimal number of significant digits, see `print.default`.
`...`	arguments passed to `print`

Details

summary Gives the details of the starship.adaptivegrid and optim steps.

Author(s)

Robert King, [email protected], https://github.com/newystats/

Darren Wraith

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.

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

King, R.A.R. & MacGillivray, H. L. (1999), A starship method for fitting the generalised $\lambda$ distributions, Australian and New Zealand Journal of Statistics 41, 353–374

Owen, D. B. (1988), The starship, Communications in Statistics - Computation and Simulation 17, 315–323.

https://github.com/newystats/gld/

Examples

data <- rgl(100,0,1,.2,.2)
starship.result <- starship(data,optim.method="Nelder-Mead",initgrid=list(lcvect=(0:4)/10,
ldvect=(0:4)/10))
print(starship.result)
summary(starship.result,estimation.details=TRUE)
data <- rgl(100,0,1,.2,.2)
starship.result <- starship(data,optim.method="Nelder-Mead",initgrid=list(lcvect=(0:4)/10,
ldvect=(0:4)/10))
print(starship.result)
summary(starship.result,estimation.details=TRUE)

Deprecated function for density quantile function of gld. See qdgl instead

Description

See qdgl help instead.

Usage

qdgl(p, lambda1, lambda2 = NULL, lambda3 = NULL, lambda4 = NULL, 
  param = "fkml", lambda5 = NULL)
qdgl(p, lambda1, lambda2 = NULL, lambda3 = NULL, lambda4 = NULL, 
  param = "fkml", lambda5 = NULL)

Arguments

`p`	See qdgl help instead.
`lambda1`	See qdgl help instead.
`lambda2`	See qdgl help instead.
`lambda3`	See qdgl help instead.
`lambda4`	See qdgl help instead.
`param`	See qdgl help instead.
`lambda5`	See qdgl help instead.

Value

See qdgl help instead.

Quantile-Quantile plot against the generalised lambda distribution

Description

qqgl produces a Quantile-Quantile plot of data against the generalised lambda distribution, or a Q-Q plot to compare two sets of parameter values for the generalised lambda distribution. It does for the generalised lambda distribution what qqnorm does for the normal.

Usage

qqgl(y = NULL, lambda1 = 0, lambda2 = NULL, lambda3 = NULL, lambda4 = NULL, 
  param = "fkml", lambda5 = NULL, abline = TRUE, lambda.pars1 = NULL, lambda.pars2 = NULL,
  param2 = "fkml", points.for.2.param.sets = 4000, ...)
qqgl(y = NULL, lambda1 = 0, lambda2 = NULL, lambda3 = NULL, lambda4 = NULL, 
  param = "fkml", lambda5 = NULL, abline = TRUE, lambda.pars1 = NULL, lambda.pars2 = NULL,
  param2 = "fkml", points.for.2.param.sets = 4000, ...)

Arguments

`y`	The data sample
`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. Alternatively, leave `lambda1` as the default value of 0 and use the `lambda.pars1` argument instead. 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)
`abline`	A logical value, TRUE adds a line through the origian with a slope of 1 to the plot
`lambda.pars1`	Parameters of the generalised lambda distribution (see `lambda1` to `lambda4` for details.
`lambda.pars2`	Second set of parameters of the generalised lambda distribution (see `lambda1` to `lambda4` for details. Use `lambda.pars1` and `lambda.pars2` to produce a QQ plot comparing two generalised lambda distributions
`param2`	parameterisation to use for the second set of parameter values
`points.for.2.param.sets`	Number of quantiles to use in a Q-Q plot comparing two sets of parameter values
`...`	graphical parameters, passed to `qqplot`

Details

See gld for more details on the Generalised Lambda Distribution. A Q-Q plot provides a way to visually assess the correspondence between a dataset and a particular distribution, or between two distributions.

Value

A list of the same form as that returned by qqline

`x`	The x coordinates of the points that were/would be plotted, corresponding to a generalised lambda distibution with parameters $\lambda_1$ , $\lambda_2$ , $\lambda_3$ , $\lambda_4$ .
`y`	The original `y` vector, i.e., the corresponding y coordinates, or a corresponding set of quantiles from a generalised lambda distribution with the second set of parameters

Author(s)

Robert King, [email protected], https://github.com/newystats/

References

King, R.A.R. & MacGillivray, H. L. (1999), A starship method for fitting the generalised $\lambda$ distributions, Australian and New Zealand Journal of Statistics 41, 353–374

https://github.com/newystats/gld/

Examples

qqgl(rgl(100,0,1,0,-.1),0,1,0,-.1)
qqgl(lambda1=c(0,1,0.01,0.01),lambda.pars2=c(0,.01,0.01,0.01),param2="rs",pch=".")
qqgl(rgl(100,0,1,0,-.1),0,1,0,-.1)
qqgl(lambda1=c(0,1,0.01,0.01),lambda.pars2=c(0,.01,0.01,0.01),param2="rs",pch=".")

Carry out the “starship” estimation method for the generalised lambda distribution

Description

Estimates parameters of the generalised lambda distribution on the basis of data, using the starship method. The starship method is built on the fact that the generalised lambda distribution (gld) is a transformation of the uniform distribution. This method finds the parameters that transform the data closest to the uniform distribution. This function uses a grid-based search to find a suitable starting point (using starship.adaptivegrid) then uses optim to find the parameters that do this.

Usage

starship(data, optim.method = "Nelder-Mead", initgrid = NULL, 
inverse.eps = .Machine$double.eps, param="FMKL", optim.control=NULL, return.data=FALSE)
starship(data, optim.method = "Nelder-Mead", initgrid = NULL, 
inverse.eps = .Machine$double.eps, param="FMKL", optim.control=NULL, return.data=FALSE)

Arguments

data

Data to be fitted, as a vector

optim.method

Optimisation method for optim to use, defaults to Nelder-Mead

initgrid

Grid of values of $\lambda_3$ and $\lambda_4$ to try, in starship.adaptivegrid. This should be a list with elements, lcvect, a vector of values for $\lambda_3$ , ldvect, a vector of values for $\lambda_4$ and levect, a vector of values for $\lambda_5$ (levect is only required if param is fm5).

If it is left as NULL, the default grid depends on the parameterisation. For fmkl, both lcvect and ldvect default to:

-1.5

-1

-0.5

-0.1

0.1

0.2

0.4

0.8

1.5

(levect is NULL).

For rs, both lcvect and ldvect default to:

0.1

0.2

0.4

0.8

1.5

(levect is NULL). Note that this restricts the estimates to only part of the region of the $\lambda_3$ , $\lambda_4$ plane.

For gpd, the defaults are: $\delta$ :

0.3

0.5

0.7

and $\lambda$ :

-1.5

-.5

0.8

1.5

For fm5, both lcvect and ldvect default to:

-1.5

-1

-.5

-0.1

0.1

0.2

0.4

0.8

1.5

and levect defaults to:

-0.5

0.25

0.5

inverse.eps

Accuracy of calculation for the numerical determination of $F(x)$ , defaults to .Machine$double.eps

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)

optim.control

List of options for the optimisation step. See optim for details. If left as NULL, the parscale control is set to scale $\lambda_1$ and $\lambda_2$ by the absolute value of their starting points.

return.data

Logical: Should the function return the data (from the argument data)?

Details

The starship method is described in King & MacGillivray, 1999 (see references). It is built on the fact that the generalised lambda distribution (gld) is a transformation of the uniform distribution. Thus the inverse of this transformation is the distribution function for the gld. The starship method applies different values of the parameters of the distribution to the distribution function, calculates the depths q corresponding to the data and chooses the parameters that make the depths closest to a uniform distribution.

The closeness to the uniform is assessed by calculating the Anderson-Darling goodness-of-fit test on the transformed data against the uniform, for a sample of size length(data).

This is implemented in 2 stages in this function. First a grid search is carried out, over a small number of possible parameter values (see starship.adaptivegrid for details). Then the minimum from this search is given as a starting point for an optimisation of the Anderson-Darling value using optim, with method given by optim.method

See GeneralisedLambdaDistribution for details on parameterisations.

Value

starship returns an object of class "starship".

An object of class "starship" is a list containing at least the following components:

lambda

A vector of length 4 (or 5, for the fm5 parameterisation), giving the estimated parameters, in order, $\lambda_1$ - location parameter $\lambda_2$ - scale parameter $\lambda_3$ - first shape parameter $\lambda_4$ - second shape parameter (See gld for details of the parameters in the fm5 parameterisation)

In the gpd parameterisation, the parameters are labelled: $\alpha$ - location parameter $\beta$ - scale parameter $\delta$ - skewness parameter $\lambda$ - tailweight parameter

grid.results

output from the grid search - see starship.adaptivegrid for details

optim

output from the optim search - optim for details

Author(s)

Robert King, [email protected], https://github.com/newystats/

Darren Wraith

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.

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

King, R.A.R. & MacGillivray, H. L. (1999), A starship method for fitting the generalised $\lambda$ distributions, Australian and New Zealand Journal of Statistics 41, 353–374

Owen, D. B. (1988), The starship, Communications in Statistics - Computation and Simulation 17, 315–323.

https://github.com/newystats/gld/

Examples

exampledata <- rgl(300,c(0,1,0.2,0))
starship(exampledata)
exampledata <- rgl(300,c(0,1,0.2,0))
starship(exampledata)

Carry out the “starship” estimation method for the generalised lambda distribution using a grid-based search

Description

Calculates estimates for the generalised lambda distribution on the basis of data, using the starship method. The starship method is built on the fact that the generalised lambda distribution (gld) is a transformation of the uniform distribution. This method finds the parameters that transform the data closest to the uniform distribution. This function uses a grid-based search.

Usage

starship.adaptivegrid(data, initgrid,inverse.eps = 1e-08, param="FMKL")
starship.adaptivegrid(data, initgrid,inverse.eps = 1e-08, param="FMKL")

Arguments

`data`	Data to be fitted, as a vector
`initgrid`	A list with elements, `lcvect`, a vector of values for $\lambda_3$ , `ldvect`, a vector of values for $\lambda_4$ and `levect`, a vector of values for $\lambda_5$ (`levect` is only required if `param` is `fm5`). The parameter values given in `initgrid` are not checked with `gl.check.lambda`.
`inverse.eps`	Accuracy of calculation for the numerical determination of $F(x)$ , defaults to $10^{-8}$
`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)

Details

The starship method is described in King and MacGillivray, 1999 (see references). It is built on the fact that the generalised lambda distribution (gld) is a transformation of the uniform distribution. Thus the inverse of this transformation is the distribution function for the gld. The starship method applies different values of the parameters of the distribution to the distribution function, calculates the depths q corresponding to the data and chooses the parameters that make the depths closest to a uniform distribution.

The closeness to the uniform is assessed by calculating the Anderson-Darling goodness-of-fit test on the transformed data against the uniform, for a sample of size length(data).

This function carries out a grid-based search. This was the original method of King and MacGillivray, 1999, but you are advised to instead use starship which uses a grid-based search together with an optimisation based search.

See GeneralisedLambdaDistribution for details on parameterisations.

Value

`response`	The minimum “response value” — the result of the internal goodness-of-fit measure. This is the return value of starship.obj. See King and MacGillivray, 1999 for more details
`lambda`	A vector of length 4 giving the values of $\lambda_1$ to $\lambda_4$ that produce this minimum response, i.e. the estimates

Author(s)

Robert King, [email protected], https://github.com/newystats/

Darren Wraith

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.

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

King, R.A.R. & MacGillivray, H. L. (1999), A starship method for fitting the generalised $\lambda$ distributions, Australian and New Zealand Journal of Statistics 41, 353–374

Owen, D. B. (1988), The starship, Communications in Statistics - Computation and Simulation 17, 315–323.

https://github.com/newystats/gld/

Examples

data <- rgl(100,0,1,.2,.2)
starship.adaptivegrid(data,list(lcvect=(0:4)/10,ldvect=(0:4)/10))
data <- rgl(100,0,1,.2,.2)
starship.adaptivegrid(data,list(lcvect=(0:4)/10,ldvect=(0:4)/10))

Objective function that is minimised in starship estimation method

Description

The starship is a method for fitting the generalised lambda distribution. See starship for more details.

This function is the objective funciton minimised in the methods. It is a goodness of fit measure carried out on the depths of the data.

Usage

starship.obj(par, data, inverse.eps, param = "fmkl")
starship.obj(par, data, inverse.eps, param = "fmkl")

Arguments

`par`	parameters of the generalised lambda distribution, a vector of length 4, giving $\lambda_1$ to $\lambda_4$ . See `GeneralisedLambdaDistribution` for details on the definitions of these parameters
`data`	Data — a vector
`inverse.eps`	Accuracy of calculation for the numerical determination of $F(x)$ , defaults to $10^{-8}$
`param`	choose parameterisation: `fmkl` uses Freimer, Mudholkar, Kollia and Lin (1988) (default). `rs` uses Ramberg and Schmeiser (1974)

Details

The closeness to the uniform is assessed by calculating the Anderson-Darling goodness-of-fit test on the transformed data against the uniform, for a sample of size length(data).

This function returns that objective function. It is provided as a seperate function to allow users to carry out minimisations using optim or other methods. The recommended method is to use the starship function.

Value

The Anderson-Darling goodness of fit measure, computed on the transformed data, compared to a uniform distribution. Note that this is NOT the goodness-of-fit measure of the generalised lambda distribution with the given parameter values to the data.

Author(s)

Robert King, [email protected], https://github.com/newystats/ Darren Wraith

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.

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

King, R.A.R. & MacGillivray, H. L. (1999), A starship method for fitting the generalised $\lambda$ distributions, Australian and New Zealand Journal of Statistics 41, 353–374

Owen, D. B. (1988), The starship, Communications in Statistics - Computation and Simulation 17, 315–323.

https://github.com/newystats/gld/

Examples

data <- rgl(100,0,1,.2,.2)
starship.obj(c(0,1,.2,.2),data,inverse.eps=1e-10,"fmkl")
data <- rgl(100,0,1,.2,.2)
starship.obj(c(0,1,.2,.2),data,inverse.eps=1e-10,"fmkl")

Package 'gld'

Help Index

Calculates Beta function for two identical parameters, allowing non-integer negative values

Description

Usage

Arguments

Details

Value

Author(s)

References

See Also

Examples

Estimate parameters of the FKML parameterisation of the generalised lambda distribution

Description

Usage

Arguments

Details

Value

Author(s)

References

See Also

Examples

Method of moments estimation for the FKML type of the generalised lambda distribution

Description

Usage

Arguments

Details

Value

Author(s)

References

See Also

Examples

Estimate parameters of the GPD type generalised lambda distribution

Description

Usage

Arguments

Details

Value

Author(s)

References

See Also

Examples

The Generalised Lambda Distribution

Description

Usage

Arguments

Details

Value

Author(s)

References

Examples

Function to check the validity of parameters of the generalized lambda distribution

Description

Usage

Arguments

Details

Value

Note

Author(s)

References

See Also

Examples

Deprecated functions

Description

Calculate L-Moments of the GPD type generalised lambda distribution for given parameter values

Description

Usage

Arguments

Details

Value

Author(s)

References

See Also

Examples

Calculate moments of the FKML type of the generalised lambda distribution for given parameter values

Description

Usage

Arguments

Details

Value