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.
pargld(lmom, verbose=FALSE, initkh=NULL, eps=1e-3,
aux=c("tau5", "tau6"), checklmom=TRUE, ...)An R
list is returned if result='best'.
The type of distribution: gld.
The parameters of the distribution.
Difference between the \(\tilde{\tau}_5\) of the fitted distribution and true \(\hat{\tau}_5\).
Smallest sum of square error found.
The source of the parameters: “pargld”.
An R data.frame of other solutions if found.
The rest of the solutions have the following:
The location parameter of the distribution.
The scale parameter of the distribution.
The 1st shape parameter of the distribution.
The 2nd shape parameter of the distribution.
The attempt number that found valid TL-moments and parameters of GLD.
The absolute difference between \(\hat{\tau}^{(1)}_5\) of data to \(\tilde{\tau}^{(1)}_5\) of the fitted distribution.
The sum of square error found.
The starting point of the \(\kappa\) parameter.
The starting point of the \(h\) parameter.
Logical on validity of the GLD---TRUE by this point.
Logical on validity of the L-moments---TRUE by this point.
Logical on whether error was less than eps---TRUE by this point.
An L-moment object created by lmoms, vec2lmom, or TLmoms with trim=0.
A logical switch on the verbosity of output. Default is verbose=FALSE.
A vector of the initial guess of the \(\kappa\) and \(h\) parameters. No other regions of parameter space are consulted.
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.
Control the algorithm to order solutions based on smallest error in \(\Delta \tau_5\) or \(\Delta \tau_6\).
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.
W.H. Asquith
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.
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.
lmomgld, cdfgld, pdfgld, quagld, parTLgld