frechet function

Frechet distribution

Computes the pdf, cdf, value at risk and expected shortfall for the Fr'echet distribution due to Fr'echet (1927) given by [REMOVE_ME]$\begin{array}{ll}&\displaystylef (x) = \frac {\alpha \sigma^\alpha}{x^{\alpha + 1}} \exp \left\{ -\left( \frac {\sigma}{x} \right)^{\alpha} \right\},\\&\displaystyleF (x) = \exp \left\{ -\left( \frac {\sigma}{x} \right)^{\alpha} \right\},\\&\displaystyle{\rm VaR}_p (X) = \sigma \left[ -\log p \right]^{-1 / \alpha},\\&\displaystyle{\rm ES}_p (X) = \frac {\sigma}{p} \Gamma \left( 1 - 1 / \alpha, -\log p \right)\end{array} [REMOVE_ME_2]$

for $x > 0$, $0 < p < 1$, $\alpha > 0$, the shape parameter, and $\sigma > 0$, the scale parameter, where $\Gamma (a, x) = \int_x^\infty t^{a - 1} \exp \left( -t \right) dt$

denotes the complementary incomplete gamma function.

Description

Computes the pdf, cdf, value at risk and expected shortfall for the Fr'echet distribution due to Fr'echet (1927) given by

for $x > 0$, $0 < p < 1$, $\alpha > 0$, the shape parameter, and $\sigma > 0$, the scale parameter, where $\Gamma (a, x) = \int_x^\infty t^{a - 1} \exp \left( -t \right) dt$

denotes the complementary incomplete gamma function.

dfrechet(x, alpha=1, sigma=1, log=FALSE)
pfrechet(x, alpha=1, sigma=1, log.p=FALSE, lower.tail=TRUE)
varfrechet(p, alpha=1, sigma=1, log.p=FALSE, lower.tail=TRUE)
esfrechet(p, alpha=1, sigma=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
• sigma: the value of the scale parameter, must be positive, the default is 1
• alpha: 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

Returns

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.

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, tools:::Rd_expr_doi("10.1080/03610918.2014.944658")

Author(s)

Saralees Nadarajah

Examples

x=runif(10,min=0,max=1)
dfrechet(x)
pfrechet(x)
varfrechet(x)
esfrechet(x)