gumbel2 {VaRES}

Gumbel II distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the Gumbel II distribution

\begin{array}{ll} &\displaystyle f (x) = a b x^{-a - 1} \exp \left( -b x^{-a} \right), \\ &\displaystyle F (x) = 1 - \exp \left( -b x^{-a} \right), \\ &\displaystyle {\rm VaR}_p (X) = b^{1 / a} \left[ -\log (1 - p) \right]^{-1 / a}, \\ &\displaystyle {\rm ES}_p (X) = \frac {b^{1 / a}}{p} \int_0^p \left[ -\log (1 - v) \right]^{-1 / a} dv \end{array}

for x > 0, 0 < p < 1, a > 0, the shape parameter, and b > 0, the scale parameter.

Usage

dgumbel2(x, a=1, b=1, log=FALSE)
pgumbel2(x, a=1, b=1, log.p=FALSE, lower.tail=TRUE)
vargumbel2(p, a=1, b=1, log.p=FALSE, lower.tail=TRUE)
esgumbel2(p, a=1, b=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

a

the value of the scale parameter, must be positive, the default is 1

b

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)
dgumbel2(x)
pgumbel2(x)
vargumbel2(x)
esgumbel2(x)