quawei: Quantile Function of the Weibull Distribution

Description

This function computes the quantiles of the Weibull distribution given parameters ($\zeta$, $\beta$, and $\delta$) computed by parwei. The quantile function is $$x(F) = \beta[- \log(1-F)]^{1/\delta} - \zeta \mbox{,}$$ where $x(F)$ is the quantile for nonexceedance probability $F$, $\zeta$ is a location parameter, $\beta$ is a scale parameter, and $\delta$ is a shape parameter.

The Weibull distribution is a reverse Generalized Extreme Value distribution. As result, the Generalized Extreme Value algorithms are used for implementation of the Weibull in lmomco. The relations between the Generalized Extreme Value distribution parameters ($\xi$, $\alpha$, $\kappa$) are $\kappa$) is $\kappa = 1/\delta$, $\alpha = \beta/\delta$, and $\xi = \zeta - \beta$. These relations are taken from Hosking and Wallis (1997).

In R, the quantile function of the Weibull distribution is qweibull. Given a Weibull parameter object p, the R syntax is qweibull(f, p$para[3], scale=p$para[2]) - p$para[1]. For the current implementation for this package, the reversed Generalized Extreme Value distribution quagev is used and the implementation is -quagev((1-f),para).

Usage

quawei(f, para, paracheck=TRUE)

Value

Quantile value for nonexceedance probability $F$.

Arguments

f: Nonexceedance probability ($0 \le F \le 1$).
para: The parameters from parwei 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.

Author

W.H. Asquith

References

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

Examples

Run this code

  # Evaluate Weibull deployed here and within R (qweibull)
  lmr <- lmoms(c(123,34,4,654,37,78))
  WEI <- parwei(lmr)
  Q1  <- quawei(0.5,WEI)
  Q2  <- qweibull(0.5,shape=WEI$para[3],scale=WEI$para[2])-WEI$para[1]
  if(Q1 == Q2) EQUAL <- TRUE

  # The Weibull is a reversed generalized extreme value
  Q <- sort(rlmomco(34,WEI)) # generate Weibull sample
  lm1 <- lmoms(Q)    # regular L-moments
  lm2 <- lmoms(-Q)   # L-moment of negated (reversed) data
  WEI <- parwei(lm1) # parameters of Weibull
  GEV <- pargev(lm2) # parameters of GEV
  F <- nonexceeds()  # Get a vector of nonexceedance probs
  plot(pp(Q),Q)
  lines(F,quawei(F,WEI))
  lines(F,-quagev(1-F,GEV),col=2) # line over laps previous

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