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

pargld {lmomco}

R Documentation

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.

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)

[Package lmomco version 2.5.1 Index]