Method of moments estimation for the FKML type of the generalised lambda distribution using given moment values
Estimates parameters of the generalised lambda distribution (FKML type) using the Method of Moments on the basis of moment values (mean, variance, skewness ratio and kurtosis ratio (note, not the excess kurtosis)).
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))
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.control |
argument |
starting.point |
a vector of length 2, giving the starting value for lambda 3 and lambda 4. |
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 (to come in version 2.4 of the gld package) will estimate
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
lambda 2 and then 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.
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)
Sigbert Klinke
Paul van Staden
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., 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
# 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)) # 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)
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