# quawei: Quantile Function of the Weibull Distribution In lmomco: L-Moments, Censored L-Moments, Trimmed L-Moments, L-Comoments, and Many Distributions

 quawei R Documentation

## Quantile Function of the Weibull Distribution

### Description

This function computes the quantiles of the Weibull distribution given parameters (ζ, β, and δ) computed by `parwei`. The quantile function is

x(F) = β[- \log(1-F)]^{1/δ} - ζ \mbox{,}

where x(F) is the quantile for nonexceedance probability F, ζ is a location parameter, β is a scale parameter, and δ 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 (ξ, α, κ) are κ) is κ = 1/δ, α = β/δ, and ξ = ζ - β. 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, scale=p\$para) - p\$para`. 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)
```

### Arguments

 `f` Nonexceedance probability (0 ≤ F ≤ 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.

### Value

Quantile value for nonexceedance probability F.

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.

`cdfwei`, `pdfwei`, `lmomwei`, `parwei`

### Examples

```  # 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,scale=WEI\$para)-WEI\$para
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
```

lmomco documentation built on Aug. 27, 2022, 1:06 a.m.