pargld: Estimate the Parameters of the Generalized Lambda...

In wasquith/lmomco: L-Moments, Censored L-Moments, Trimmed L-Moments, L-Comoments, and Many Distributions

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, ...)

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.

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.

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.

Note

This function is a cumbersome method of parameter solution, but years of testing suggest that with supervision and the available options regarding the optimization that reliable parameter estimations result. The Tukey Lambda distribution is a special form of the GLD, see Tukey Lambda Notes section in Details of lmrdia46 for more details.

Author(s)

W.H. Asquith

Source

W.H. Asquith in Feb. 2006 with a copy of Karian and Dudewicz (2000) and again Feb. 2011.

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.

See Also

lmomgld, cdfgld, pdfgld, quagld, parTLgld

Examples

## Not run: 
  X      <- sort( rgamma(202, 2) ) # simulate a skewed distribution
  lmr    <- lmoms(X)               # compute trimmed L-moments
  PARgld <- pargld(lmr)            # fit the GLD
  FF     <- pp(X)
  plot( FF,    X, col=8, cex=0.25)
  lines(FF, qlmomco(FF, 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(FF, qlmomco(FF, vec2par(other, type="gld")), col="red")
  }
  # 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) # 
## End(Not run)

## Not run: 
  FF <- seq(0.01, 0.99, 0.01)
  plot(FF,  qlmomco(FF, vec2par(c(3.1446434, 2.943469, 7.4211316, 1.050537),
                                type="gld")), col="blue", type="l")
  lines(FF, qlmomco(FF, vec2par(c(0.4962471, 8.794038, 0.0082958, 0.228352),
                                type="gld")), col="red"           ) # 
## End(Not run)

wasquith/lmomco documentation built on Nov. 13, 2024, 4:53 p.m.