parTLgld: Estimate the Parameters of the Generalized Lambda Distribution using Trimmed L-moments (t=1)

Description

This function estimates the parameters of the Generalized Lambda distribution given the trimmed L-moments (TL-moments) for $t=1$ of the data in a TL-moment object with a trim level of unity (trim=1). The relation between distribution parameters and TL-moments is seen under lmomTLgld. There are no simple expressions for the parameters in terms of the L-moments. This function is considered EXPERIMENTAL and general details of the algorithm are provided below. Further, consider that multiple parameter solutions are possible with the Generalized Lambda so some expertise in the distribution and other aspects are needed.

Usage

parTLgld(lmom,result='best',verbose=FALSE,extract=0,initkh=NULL)

Arguments

lmom

A TL-moment object created by TLmoms.

result

If best, then the minimum error solution is returned. If dataframe, then data.frame is returned with sequence of valid solutions sorted in ascending error order.

verbose

A logical switch on the verbosity of output. Default is verbose=FALSE.

extract

If result=dataframe and extract greater than zero, then the extract=n returns the nth element of the data.frame as if that element was the best solution.

initkh

A vector of the initial guess of the $\kappa$ and $h$ parameters. No other regions of parameter space are consulted.

Value

An R list is returned if result='best'.
typeThe type of distribution: gld.
paraThe parameters of the distribution.
errorSmallest sum of square error found.
tau5diffDifference between true $\hat{\tau}^{(1)}_5$ and the $\tilde{\tau}^{(1)}_5$ of the fitted distribution.
sourceThe source of the parameters: parTLgld.
An R data.frame is returned if result='dataframe', which is sorted by ascending error.
attemptThe attempt number that found valid TL-moments and parameters of GLD.
xThe location parameter of the distribution.
aThe scale parameter of the distribution.
kThe 1st shape parameter of the distribution.
hThe 2nd shape parameter of the distribution.
tau5_diffThe absolute difference between $\hat{\tau}^{(1)}_5$ of data to $\tilde{\tau}^{(1)}_5$ of the fitted distribution.
errorThe sum of square error found.
initial_kThe starting point of the $\kappa$ parameter.
initial_hThe starting point of the $h$ parameter.

source

R hacking by W.H. Asquith in February 2006 with copy of Karian and Dudewicz (2000).

Details

Karian and Dudewicz (2000) summarize six regions of the $\kappa$ and $h$ space in which the Generalized Lambda distribution is valid for suitably choosen $\alpha$. Numerical experimentation suggestions that the L-moments are not valid in Regions 1 and 2. However, initial guesses of the parameters within each region are used numerous six separate optim (the R function) efforts to perform a least sum-of-square errors on the following objective function.

$$(\hat{\tau}^{(1)}_3 - \tilde{\tau}^{(1)}_3)^2 + (\hat{\tau}^{(1)}_4 - \tilde{\tau}^{(1)}_4)^2 \mbox{, }$$

where $\tilde{\tau}^{(1)}_r$ is the L-moment ratio of the data, $\hat{\tau}^{(1)}_r$ is the estimated value of the TL-moment ratio for the current pairing of $\kappa$ and $h$ and $\tau^{(1)}_r$ is the actual value of the L-moment ratio.

For each optimization a check on the validity of the parameters so produced is made--are the parameters consistent with the Generalized Lambda distribution and a second check is made on the validity of $\tau^{(1)}_3$ and $\tau^{(1)}_4$. If both validity checks return TRUE then the optimization is retained if its sum-of-square error is less than the previous optimum value. It is possible for a given solution to be found outside the starting region of the initial guesses. The surface generated by the $\tau^{(1)}_3$ and $\tau^{(1)}_4$ equations seen in lmomTLgld is complex--different initial guesses within a given region can yield what appear to be radically different $\kappa$ and $h$. Users are encouraged to play with alternative solutions (see the verbose argument). A quick double check on the L-moments (not TL-moments) from the solved parameters using lmomTLgld is encouraged as well.

References

Karian, Z.A., and Dudewicz, E.J., 2000, Fitting statistical distributions---The generalized lambda distribution and generalized bootstrap methods: CRC Press, Boca Raton, FL, 438 p.

Examples

Run this code

TL1 <- TLmoms(rnorm(200),trim=1)
p <- parTLgld(TL1)
TL2 <- lmomTLgld(p)

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