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

parTLgld {lmomco}

R Documentation

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 relations between distribution parameters and TL-moments are seen under lmomTLgld. 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 distribution so some expertise with this distribution and other aspects is advised.

Usage

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

Arguments

`lmom`	A TL-moment object created by `TLmoms`.
`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 trimmed `\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}^{(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.

Value

An R list is returned if result='best'.

`type`	The type of distribution: `gld`.
`para`	The parameters of the distribution.
`delTau5`	Difference between `\tilde{\tau}^{(1)}_5` of the fitted distribution and true `\hat{\tau}^{(1)}_5`.
`error`	Smallest sum of square error found.
`source`	The source of the parameters: “parTLgld”.
`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.

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.

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

# As of version 1.6.2, it is felt that in spirit of CRAN CPU
# reduction that the intensive operations of parTLgld() should
# be kept a bay.

## Not run: 
X <- rgamma(202,2) # simulate a skewed distribution
lmr <- TLmoms(X, trim=1) # compute trimmed L-moments
PARgldTL <- parTLgld(lmr) # fit the GLD

F <- pp(X) # plotting positions for graphing
plot(F,sort(X), col=8, cex=0.25)
lines(F, qlmomco(F,PARgldTL)) # show the best estimate
if(! is.null(PARgldTL$rest)) { 
  n <- length(PARgldTL$rest$xi)
  other <- unlist(PARgldTL$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(PARgldTL)
#$type
#[1] "gld"
#$para
#         xi       alpha       kappa           h
# 1.02333964 -3.86037875 -0.06696388 -0.22100601
#$delTau5
#[1] -0.02299319
#$error
#[1] 7.048409e-08
#$source
#[1] "pargld"
#$rest
#         xi     alpha       kappa          h attempt     delTau5        error
#1  1.020725 -3.897500 -0.06606563 -0.2195527       6 -0.02302222 1.333402e-08
#2  1.021203 -3.895334 -0.06616654 -0.2196020       4 -0.02304333 8.663930e-11
#3  1.020684 -3.904782 -0.06596204 -0.2192197       5 -0.02306065 3.908918e-09
#4  1.019795 -3.917609 -0.06565792 -0.2187232       2 -0.02307092 2.968498e-08
#5  1.023654 -3.883944 -0.06668986 -0.2198679       7 -0.02315035 2.991811e-07
#6 -4.707935 -5.044057  5.89280906 -0.3261837      13  0.04168800 2.229672e-10

## End(Not run)

## Not run: 
F <- seq(.01,.99,.01)
plot(F,qlmomco(F, vec2par(c( 1.02333964, -3.86037875,
                            -0.06696388, -0.22100601), type="gld")),
     type="l")
lines(F,qlmomco(F, vec2par(c(-4.707935, -5.044057,
                              5.89280906, -0.3261837), type="gld")))

## End(Not run)

[Package lmomco version 2.5.1 Index]