quape3: Quantile Function of the Pearson Type III Distribution

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

Quantile Function of the Pearson Type III Distribution

Description

This function computes the quantiles of the Pearson Type III distribution given parameters (\mu, \sigma, and \gamma) computed by parpe3. The quantile function has no explicit form (see cdfpe3).

For the implementation in the lmomco package, the three parameters are \mu, \sigma, and \gamma for the mean, standard deviation, and skew, respectively. Therefore, the Pearson Type III distribution is of considerable theoretical interest to this package because the parameters, which are estimated via the L-moments, are in fact the product moments, although, the values fitted by the method of L-moments will not be numerically equal to the sample product moments. Further details are provided in the Examples section under pmoms.

Usage

quape3(f, para, paracheck=TRUE)

Arguments

f	Nonexceedance probability (0 \le F \le 1).
para	The parameters from parpe3 or vec2par.
paracheck	A logical controlling whether the parameters are checked for validity. Overriding of this check might be extremely important and needed for use of the quantile function in the context of TL-moments with nonzero trimming.

Value

Quantile value for nonexceedance probability F.

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

cdfpe3, pdfpe3, lmompe3, parpe3

Examples

  lmr <- lmoms(c(123,34,4,654,37,78))
  quape3(0.5,parpe3(lmr))

## Not run: 
  # Let us run an experiment on the reflection symmetric PE3.
  # Pick some parameters suitable for hydrologic applications in log.
  para_neg <- vec2par(c(3,.3,-1), type="pe3") # Notice only the
  para_pos <- vec2par(c(3,.3,+1), type="pe3") # sign change of skew.

  nsim <- 1000 # Number of simulations
  nsam <- 70   # Reasonable sample size in hydrology
  neg <- pos <- rep(NA, nsim)
  for(i in 1:nsim) {
    ff <- runif(nsam) # Ensure that each qlmomco()-->quape3() has same probs.
    neg[i] <- lmoms.cov(qlmomco(ff, para_neg), nmom=3, se="lmrse")[3]
    pos[i] <- lmoms.cov(qlmomco(ff, para_pos), nmom=3, se="lmrse")[3]
    # We have extracted the sample standard error of L-skew from the sample
    # This is not the same as the standard error of so computed PE3 
    # parameters, but for the illustration here, it does not matter much.
  }
  zz <- data.frame(setau3=c(neg,pos), # preserve to make grouping boxplot
                   sign=c(rep("negskew", nsim), rep("posskew", nsim)))
  boxplot(zz$setau3~zz$sign, xlab="Sign of a '1' PE3 skew",
                             ylab="Standard error of L-skew")
  mtext("Standard Errors of 1,000 PE3 Parents (3,0.3,+/-1) (n=70)")
  # Notice that the distribution of the standard errors of L-skew are 
  # basically the same whether or no the sign of the skew is reversed.
  # Finally, we make a scatter plot as a check that for any given sample
  # derived from same probabilities that the standard errors are indeed,
  # that is, remain sample specific.
  plot(neg, pos, xlab="Standard error of -1 skew simulation",
                 ylab="Standard error of +1 skew simulation")
  mtext("Standard Errors of 1,000 PE3 Parents (3,0.3,+/-1) (n=70)") # 
## End(Not run)

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