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

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

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.

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/

Author

Robert King, robert.king.newcastle@gmail.com, https://github.com/newystats/

Sigbert Klinke

Paul van Staden

Examples

gld.moments(c(0,1.463551,0.1349124,0.1349124))
#>     mean  sigmasq       a3       a4 
#> 0.000000 1.000001 0.000000 3.000000 
gld.moments(c(0,1.813799,0,0))
#>    mean sigmasq      a3      a4 
#>     0.0     1.0     0.0     4.2 
gld.moments(c(0,1,0,3))
#>      mean   sigmasq        a3        a4 
#> -0.750000  1.189484 -1.873833  7.991101