# cdfgev: Generalized extreme-value distribution In lmom: L-Moments

 cdfgev R Documentation

## Generalized extreme-value distribution

### Description

Distribution function and quantile function of the generalized extreme-value distribution.

### Usage

cdfgev(x, para = c(0, 1, 0))
quagev(f, para = c(0, 1, 0))


### Arguments

 x Vector of quantiles. f Vector of probabilities. para Numeric vector containing the parameters of the distribution, in the order \xi, \alpha, k (location, scale, shape).

### Details

The generalized extreme-value distribution with location parameter \xi, scale parameter \alpha and shape parameter k has distribution function

F(x)=\exp\lbrace-\exp(-y)\rbrace

where

y=-k^{-1}\log\lbrace1-k(x-\xi)/\alpha\rbrace,

with x bounded by \xi+\alpha/k from below if k<0 and from above if k>0, and quantile function

x(F)=\xi+{\alpha\over k}\lbrace1-(-\log F)^k\rbrace.

Extreme-value distribution types I, II and III (Gumbel, Frechet, Weibull) correspond to shape parameter values k=0, k<0 and k>0 respectively.

### Value

cdfgev gives the distribution function; quagev gives the quantile function.

### Note

The functions expect the distribution parameters in a vector, rather than as separate arguments as in the standard R distribution functions pnorm, qnorm, etc.

Two parametrizations of the generalized extreme-value distribution are in common use. When Jenkinson (1955) introduced the distribution he wrote the distribution function in the form

F(x) = \exp [ - \lbrace 1 - k ( x - \xi ) / \alpha) \rbrace^{1/k}].

and that is the form used in R package lmom. A slight inconvenience with it is that the skewness of the distribution is a decreasing function of the shape parameter k. Perhaps for this reason, authors of some other R packages prefer a form in which the sign of the shape parameter k is changed and the parameters are renamed:

F(x) = \exp [ - \lbrace 1 + \xi ( x - \mu ) / \sigma) \rbrace^{-1/\xi}].

Users should be able to mix functions from packages that use either form; just be aware that the sign of the shape parameter will need to be changed when converting from one form to the other (and that \xi is a location parameter in one form and a shape parameter in the other).

### References

Jenkinson, A. F. (1955). The frequency distribution of the annual maximum (or minimum) of meteorological elements. Quarterly Journal of the Royal Meteorological Society, 81, 158-171.

cdfgum for the Gumbel (extreme-value type I) distribution.

cdfkap for the kappa distribution, which generalizes the generalized extreme-value distribution.

cdfwei for the Weibull distribution,

### Examples

# Random sample from the generalized extreme-value distribution
# with parameters xi=0, alpha=1, k=-0.5.
quagev(runif(100), c(0,1,-0.5))


lmom documentation built on Aug. 29, 2023, 9:07 a.m.