parpe3: Estimate the Parameters of the Pearson Type III Distribution

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

Estimate the Parameters of the Pearson Type III Distribution

Description

This function estimates the parameters of the Pearson Type III distribution given the L-moments of the data in an L-moment object such as that returned by lmoms. The L-moments in terms of the parameters are complicated and solved numerically. For the implementation in lmomco, the three parameters are \mu, \sigma, and \gamma for the mean, standard deviation, and skew, respectively.

Remark In spring 2026, a study was conducted using inst/newcoes/makePE3coes.R of lmomco sources for computation of better coefficients used in approximation of the relation between \gamma and L-skew (\tau_3). A vector of \gamma values was made for \mu=0 and \sigma = 0, without a loss in generality, and the theoretical \tau_3 computed by theoLmoms and optim() for fitting. These new coefficients relative to those taken from Hosking (1996) were implemented in spring 2026 by new settings in the parpe3 source code. Slightly more accurate conversion of \tau_3 to \gamma was attained. The structural formulation by Hosking was retained and only the fitting coefficients altered.

Usage

parpe3(lmom, checklmom=TRUE, useHosking=TRUE, ...)

Arguments

lmom	An L-moment object created by lmoms or vec2lmom.
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.
useHosking	A logical to use the old Hosking (1996) FORTRAN coefficients or use the new optimized coefficients of Spring 2026 custom to the lmomco package.
...	Other arguments to pass.

Value

An R list is returned.

type	The type of distribution: pe3.
para	The parameters of the distribution.
source	The source of the parameters: “parpe3”.

Author(s)

W.H. Asquith

References

Hosking, J.R.M., 1990, L-moments—Analysis and estimation of distributions using linear combinations of order statistics: Journal of the Royal Statistical Society, Series B, v. 52, pp. 105–124.

Hosking, J.R.M., 1996, FORTRAN routines for use with the method of L-moments: Version 3, IBM Research Report RC20525, T.J. Watson Research Center, Yorktown Heights, New York.

Hosking, J.R.M., and Wallis, J.R., 1997, Regional frequency analysis—An approach based on L-moments: Cambridge University Press.

See Also

lmompe3, cdfpe3, pdfpe3, quape3

Examples

lmr <- lmoms(rnorm(20))
parpe3(lmr)

## Not run: 
  Tau3 <- runif(500000, min=-1, max=1)
  system.time(H <- sapply(Tau3, function(t) {
              parpe3(vec2lmom(c(0,1,t)), useHosking=TRUE )$para[3] }))
  #   user  system elapsed
  #  7.005   0.016   7.080
  system.time(W <- sapply(Tau3, function(t) {
              parpe3(vec2lmom(c(0,1,t)), useHosking=FALSE)$para[3] }))
  #   user  system elapsed
  #  7.084   0.023   7.186
  err <- H - W # error between the two methods
  plot(qnorm(pp(err)), abs(sort(err)), type="l", log="y",
       xlab="Standard normal variate of gamma parameter error",
       ylab="| Hosking 1996 minus lmomco 2026 |") #
## End(Not run)

lmomco documentation built on Oct. 18, 2026, 5:07 p.m.