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)=xα+1ασαexp{−(xσ)α},\displaystyleF(x)=exp{−(xσ)α},VaRp(X)=σ[−logp]−1/α,ESp(X)=pσΓ(1−1/α,−logp)[REMOVEME2]
for x>0, 0<p<1, α>0, the shape parameter, and σ>0, the scale parameter, where Γ(a,x)=∫x∞ta−1exp(−t)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
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")