gld (version 2.6.6)

fit.fkml.moments: 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))

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)

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 \(\lambda_4\).

Author

Robert King, , https://github.com/newystats/

Sigbert Klinke

Paul van Staden

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.

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/

See Also

gld.moments

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)