gompertz {VaRES}

Gompertz distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the Gompertz distribution due to Gompertz (1825) given by

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

for x > 0, 0 < p < 1, b > 0, the first scale parameter and \eta > 0, the second scale parameter.

Usage

dgompertz(x, b=1, eta=1, log=FALSE)
pgompertz(x, b=1, eta=1, log.p=FALSE, lower.tail=TRUE)
vargompertz(p, b=1, eta=1, log.p=FALSE, lower.tail=TRUE)
esgompertz(p, b=1, eta=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

b

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

eta

the value of the second scale 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)
dgompertz(x)
pgompertz(x)
vargompertz(x)
esgompertz(x)