# parTLgld: Estimate the Parameters of the Generalized Lambda... In lmomco: L-Moments, Censored L-Moments, Trimmed L-Moments, L-Comoments, and Many Distributions

## 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

 1 2 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 κ 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 τ_3 and τ_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 Δ τ_5 or Δ τ_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 τ_4 and τ_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 κ and h space in which the Generalized Lambda distribution is valid for suitably choosen α. 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{τ}^{(1)}_3 - \tilde{τ}^{(1)}_3)^2 + (\hat{τ}^{(1)}_4 - \tilde{τ}^{(1)}_4)^2 \mbox{, }

where \tilde{τ}^{(1)}_r is the L-moment ratio of the data, \hat{τ}^{(1)}_r is the estimated value of the TL-moment ratio for the current pairing of κ and h and τ^{(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 τ^{(1)}_3 and τ^{(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 τ^{(1)}_3 and τ^{(1)}_4 equations seen in lmomTLgld is complex; different initial guesses within a given region can yield what appear to be radically different κ 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{τ}^{(1)}_5 of the fitted distribution and true \hat{τ}^{(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{τ}^{(1)}_5 of data to \tilde{τ}^{(1)}_5 of the fitted distribution. error The sum of square error found. initial_k The starting point of the κ 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.

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.

TLmoms, lmomTLgld, cdfgld, pdfgld, quagld, pargld
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 # 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 # "gld" #$para # xi alpha kappa h # 1.02333964 -3.86037875 -0.06696388 -0.22100601 #$delTau5 # -0.02299319 #$error # 7.048409e-08 #$source # "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)