Frechet distribution

Description

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

\begin{array}{ll} &\displaystyle f (x) = \frac {α σ^α}{x^{α + 1}} \exp ≤ft\{ -≤ft( \frac {σ}{x} \right)^{α} \right\}, \\ &\displaystyle F (x) = \exp ≤ft\{ -≤ft( \frac {σ}{x} \right)^{α} \right\}, \\ &\displaystyle {\rm VaR}_p (X) = σ ≤ft[ -\log p \right]^{-1 / α}, \\ &\displaystyle {\rm ES}_p (X) = \frac {σ}{p} Γ ≤ft( 1 - 1 / α, -\log p \right) \end{array}

for x > 0, 0 < p < 1, α > 0, the shape parameter, and σ > 0, the scale parameter, where Γ (a, x) = \int_x^∞ t^{a - 1} \exp ≤ft( -t \right) dt denotes the complementary incomplete gamma function.

Usage

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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

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

S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall, submitted

Examples

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