frechet function

Frechet distribution

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]\displaystylef(x)=ασαxα+1exp{(σx)α},\displaystyleF(x)=exp{(σx)α},VaRp(X)=σ[logp]1/α,ESp(X)=σpΓ(11/α,logp)[REMOVEME2] \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>0x > 0, 0<p<10 < p < 1, α>0\alpha > 0, the shape parameter, and σ>0\sigma > 0, the scale parameter, where Γ(a,x)=xta1exp(t)dt\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

\displaystylef(x)=ασαxα+1exp{(σx)α},\displaystyleF(x)=exp{(σx)α},VaRp(X)=σ[logp]1/α,ESp(X)=σpΓ(11/α,logp) \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}

for x>0x > 0, 0<p<10 < p < 1, α>0\alpha > 0, the shape parameter, and σ>0\sigma > 0, the scale parameter, where Γ(a,x)=xta1exp(t)dt\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)
  • Maintainer: Leo Belzile
  • License: GPL (>= 2)
  • Last published: 2023-04-22

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