R: Method of moments estimation for the FKML type of the...

fit.fkml.moments.val {gld} | R Documentation

## Method of moments estimation for the FKML type of the generalised lambda distribution using given moment values

### Description

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)).

### Usage

    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

`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` (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.

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

### See Also

`gld.moments`

### Examples

    # 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)

[Package gld version 2.6.2 Index]