pargld: Estimate the Parameters of the Generalized Lambda Distribution

Description

This function estimates the parameters of the Generalized Lambda distribution given the L-moments of the data in an ordinary L-moment object (lmoms) or a trimmed L-moment object (TLmoms for t=1). The relations between distribution parameters and L-moments are seen under lmomgld. There are no simple expressions for the parameters in terms of the L-moments. Consider that multiple parameter solutions are possible with the Generalized Lambda so some expertise in the distribution and other aspects are needed.

Usage

pargld(lmom, verbose=FALSE, initkh=NULL, eps=1e-3,
       aux=c("tau5", "tau6"), checklmom=TRUE, ...)

Value

An R

list is returned if result='best'.

type: The type of distribution: gld.
para: The parameters of the distribution.
delTau5: Difference between the $\tilde{\tau}_5$ of the fitted distribution and true $\hat{\tau}_5$.
error: Smallest sum of square error found.
source: The source of the parameters: “pargld”.
rest: An R data.frame of other solutions if found.

The rest of the solutions have the following:

xi: The location parameter of the distribution.
alpha: The scale parameter of the distribution.
kappa: The 1st shape parameter of the distribution.
h: The 2nd shape parameter of the distribution.
attempt: The attempt number that found valid TL-moments and parameters of GLD.
delTau5: The absolute difference between $\hat{\tau}^{(1)}_5$ of data to $\tilde{\tau}^{(1)}_5$ of the fitted distribution.
error: The sum of square error found.
initial_k: The starting point of the $\kappa$ parameter.
initial_h: The starting point of the $h$ parameter.
valid.gld: Logical on validity of the GLD---TRUE by this point.
valid.lmr: Logical on validity of the L-moments---TRUE by this point.
lowerror: Logical on whether error was less than eps---TRUE by this point.

Arguments

lmom: An L-moment object created by lmoms, vec2lmom, or TLmoms with trim=0.
verbose: A logical switch on the verbosity of output. Default is verbose=FALSE.
initkh: A vector of the initial guess of the $\kappa$ and $h$ parameters. No other regions of parameter space are consulted.
eps: A small term or threshold for which the square root of the sum of square errors in $\tau_3$ and $\tau_4$ is compared to to judge “good enough” for the alogrithm to order solutions based on smallest error as explained in next argument.
aux: Control the algorithm to order solutions based on smallest error in $\Delta \tau_5$ or $\Delta \tau_6$.
checklmom: Should the lmom be checked for validity using the are.lmom.valid function. Normally this should be left as the default and it is very unlikely that the L-moments will not be viable (particularly in the $\tau_4$ and $\tau_3$ inequality). However, for some circumstances or large simulation exercises then one might want to bypass this check.
...: Other arguments to pass.

Author

W.H. Asquith

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 with numerous separate optim (the R function) efforts to perform a least sum-of-square errors on the following objective function $$(\hat{\tau}_3 - \tilde{\tau}_3)^2 + (\hat{\tau}_4 - \tilde{\tau}_4)^2 \mbox{, }$$ where $\hat{\tau}_r$ is the L-moment ratio of the data, $\tilde{\tau}_r$ is the estimated value of the L-moment ratio for the fitted distribution $\kappa$ and $h$ and $\tau_r$ 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? A second check is made on the validity of $\tau_3$ and $\tau_4$. 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 $\tau_3$ and $\tau_4$ equations seen in lmomgld 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 from the solved parameters using lmomgld is encouraged as well. Karvanen and others (2002, eq. 25) provide an equation expressing $\kappa$ and $h$ as equal (a symmetrical Generalized Lambda distribution) in terms of $\tau_4$ and suggest that the equation be used to determine initial values for the parameters. The Karvanen equation is used on a semi-experimental basis for the final optimization attempt by pargld.

References

Asquith, W.H., 2007, L-moments and TL-moments of the generalized lambda distribution: Computational Statistics and Data Analysis, v. 51, no. 9, pp. 4484--4496.

Karvanen, J., Eriksson, J., and Koivunen, V., 2002, Adaptive score functions for maximum likelihood ICA: Journal of VLSI Signal Processing, v. 32, pp. 82--92.

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

if (FALSE) {
X <- rgamma(202,2) # simulate a skewed distribution
lmr <- lmoms(X) # compute trimmed L-moments
PARgld <- pargld(lmr) # fit the GLD
F <- pp(X)
plot(F,sort(X), col=8, cex=0.25)
lines(F, qlmomco(F,PARgld)) # show the best estimate
if(! is.null(PARgld$rest)) { #$
  n <- length(PARgld$rest$xi)
  other <- unlist(PARgld$rest[n,1:4]) #$ # show alternative
  lines(F, qlmomco(F,vec2par(other, type="gld")), col=2)
}
# Note in the extraction of other solutions that no testing for whether
# additional solutions were found is made. Also, it is quite possible
# that the other solutions "[n,1:4]" is effectively another numerical
# convergence on the primary solution. Some users of this example thus
# might not see two separate lines. Users are encouraged to inspect the
# rest of the solutions: print(PARgld$rest); #$

# For one run of the above example, the GLD results follow
#print(PARgld)
#$type
#[1] "gld"
#$para
#      xi    alpha    kappa        h
#3.144379 2.943327 7.420334 1.050792
#$delTau5
#[1] -0.0367435
#$error
#[1] 5.448016e-10
#$source
#[1] "pargld"
#$rest
#         xi    alpha       kappa         h attempt     delTau5        error
#1 3.1446434 2.943469 7.421131671 1.0505370      14 -0.03675376 6.394270e-10
#2 0.4962471 8.794038 0.008295796 0.2283519       4 -0.04602541 8.921139e-10
}
if (FALSE) {
F <- seq(.01,.99,.01)
plot(F, qlmomco(F, vec2par(c(3.1446434, 2.943469, 7.4211316, 1.050537), type="gld")),
     type="l")
lines(F,qlmomco(F, vec2par(c(0.4962471, 8.794038, 0.0082958, 0.228352), type="gld")))
}

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