gev {VaRES} | R Documentation |
Generalized extreme value distribution
Description
Computes the pdf, cdf, value at risk and expected shortfall for the generalized extreme value distribution due to Fisher and Tippett (1928) given by
\begin{array}{ll}
&\displaystyle
f(x) =
\frac {1}{\sigma} \left[ 1 + \xi \left( \frac {x - \mu}{\sigma} \right) \right]^{-1/\xi - 1}
\exp \left\{ -\left[ 1 + \xi \left( \frac {x - \mu}{\sigma} \right)
\right]^{-1/\xi} \right\},
\\
&\displaystyle
F(x) = \exp \left\{ -\left[ 1 + \xi \left( \frac {x - \mu}{\sigma} \right) \right]^{-1/\xi} \right\},
\\
&\displaystyle
{\rm VaR}_p (X) = \mu - \frac {\sigma}{\xi} + \frac {\sigma}{\xi} (-\log p)^{-\xi},
\\
&\displaystyle
{\rm ES}_p (X) = \mu - \frac {\sigma}{\xi} + \frac {\sigma}{p \xi} \int_0^p (-\log v)^{-\xi} dv
\end{array}
for x \geq \mu - \sigma / \xi
if \xi > 0
,
x \leq \mu - \sigma / \xi
if \xi < 0
,
-\infty < x < \infty
if \xi = 0
,
0 < p < 1
, -\infty < \mu < \infty
, the location parameter,
\sigma > 0
, the scale parameter, and -\infty < \xi < \infty
, the shape parameter.
Usage
dgev(x, mu=0, sigma=1, xi=1, log=FALSE)
pgev(x, mu=0, sigma=1, xi=1, log.p=FALSE, lower.tail=TRUE)
vargev(p, mu=0, sigma=1, xi=1, log.p=FALSE, lower.tail=TRUE)
esgev(p, mu=0, sigma=1, xi=1)
Arguments
x |
scaler or vector of values at which the pdf or cdf needs to be computed |
p |
scaler or vector of values at which the value at risk or expected shortfall needs to be computed |
mu |
the value of the location parameter, can take any real value, the default is zero |
sigma |
the value of the scale parameter, must be positive, the default is 1 |
xi |
the value of the shape parameter, must be positive, the default is 1 |
log |
if TRUE then log(pdf) are returned |
log.p |
if TRUE then log(cdf) are returned and quantiles are computed for exp(p) |
lower.tail |
if FALSE then 1-cdf are returned and quantiles are computed for 1-p |
Value
An object of the same length as x
, giving the pdf or cdf values computed at x
or an object of the same length as p
, giving the values at risk or expected shortfall computed at p
.
Author(s)
Saralees Nadarajah
References
Stephen Chan, Saralees Nadarajah & Emmanuel Afuecheta (2016). An R Package for Value at Risk and Expected Shortfall, Communications in Statistics - Simulation and Computation, 45:9, 3416-3434, doi:10.1080/03610918.2014.944658
Examples
x=runif(10,min=0,max=1)
dgev(x)
pgev(x)
vargev(x)
esgev(x)