# Quantile Function of the Weibull Distribution This function computes the quantiles of the Weibull distribution given parameters ($\zeta$, $\beta$, and $\delta$) computed by `parwei`. The quantile function is [REMOVE_ME]$$ x(F) = \beta[- \log(1-F)]^{1/\delta} - \zeta \mbox{,} [REMOVE_ME_2]$$ 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 , the quantile function of the Weibull distribution is `qweibull`. Given a Weibull parameter object `p`, the syntax is `qweibull(f, p$para[3], scale=p$para[2]) - p$para[1]`. For the current implementation for this package, the reversed Generalized Extreme Value distribution `quagev` is used and the implementation is `-quagev((1-f),para)`. ## Description This function computes the quantiles of the Weibull distribution given parameters ($\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 , the quantile function of the Weibull distribution is `qweibull`. Given a Weibull parameter object `p`, the 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)`. ```r quawei(f, para, paracheck=TRUE) ``` ## 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. ## Returns Quantile value for nonexceedance probability $F$. ## References Hosking, J.R.M., and Wallis, J.R., 1997, Regional frequency analysis---An approach based on L-moments: Cambridge University Press. ## Author(s) W.H. Asquith ## See Also `cdfwei`, `pdfwei`, `lmomwei`, `parwei` ## Examples ```r # 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 ```

quawei function