# betagumbel2: Beta Gumbel 2 distribution In VaRES: Computes value at risk and expected shortfall for over 100 parametric distributions

## Description

Computes the pdf, cdf, value at risk and expected shortfall for the beta Gumbel II distribution given by

\begin{array}{ll} &\displaystyle f (x) = \frac {a b x^{-a - 1}}{B (c, d)} \exp ≤ft( -b d x^{-a} \right) ≤ft[ 1 - \exp ≤ft( -b x^{-a} \right) \right]^{c - 1}, \\ &\displaystyle F (x) = I_{1 - \exp ≤ft( -b x^{-a} \right)} (c, d), \\ &\displaystyle {\rm VaR}_p (X) = b^{1 / a} ≤ft\{ -\log ≤ft[ 1 - I_p^{-1} (c, d) \right] \right\}^{-1 / a}, \\ &\displaystyle {\rm ES}_p (X) = \frac {b^{1 / a}}{p} \int_0^p ≤ft\{ -\log ≤ft[ 1 - I_v^{-1} (c, d) \right] \right\}^{-1 / a} dv \end{array}

for x > 0, 0 < p < 1, a > 0, the first shape parameter, b > 0, the scale parameter, c > 0, the second shape parameter, and d > 0, the third shape parameter.

## Usage

 1 2 3 4 dbetagumbel2(x, a=1, b=1, c=1, d=1, log=FALSE) pbetagumbel2(x, a=1, b=1, c=1, d=1, log.p=FALSE, lower.tail=TRUE) varbetagumbel2(p, a=1, b=1, c=1, d=1, log.p=FALSE, lower.tail=TRUE) esbetagumbel2(p, a=1, b=1, c=1, d=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 scale parameter, must be positive, the default is 1 a the value of the first shape parameter, must be positive, the default is 1 c the value of the second shape parameter, must be positive, the default is 1 d the value of the third 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)

 1 2 3 4 5 x=runif(10,min=0,max=1) dbetagumbel2(x) pbetagumbel2(x) varbetagumbel2(x) #esbetagumbel2(x)