R: Calculate moments of the FKML type of the generalised lambda...

gld.moments {gld} | R Documentation

## 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)

### 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, robert.king@newcastle.edu.au, https://tolstoy.newcastle.edu.au/~rking/

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 https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.106.6130&rep=rep1&type=pdf

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://tolstoy.newcastle.edu.au/~rking/gld/

### See Also

`fit.fkml.moments.val`

### 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))

[Package gld version 2.6.2 Index]