beard function

Beard distribution

Computes the pdf, cdf, value at risk and expected shortfall for the Beard distribution due to Beard (1959) given by [REMOVE_ME]$\begin{array}{ll}&\displaystylef(x) = \frac {\displaystyle a \exp (b x) \left[ 1 + a \rho \right]^{\rho^{-1/b}}}{\displaystyle \left[ 1 + a \rho \exp (b x) \right]^{1 + \rho^{-1/b}}},\\&\displaystyleF (x) = 1 - \frac {\displaystyle \left[ 1 + a \rho \right]^{\rho^{-1/b}}}{\displaystyle \left[ 1 + a \rho \exp (b x) \right]^{\rho^{-1/b}}},\\&\displaystyle{\rm VaR}_p (X) = \frac {1}{b} \log \left[ \frac {1 + a \rho}{a \rho (1 - p)^{\rho^{1 / b}}} - \frac {1}{a \rho} \right],\\&\displaystyle{\rm ES}_p (X) =\frac {1}{p b} \int_0^p \log \left[ -\frac {1}{a \rho} +\frac {1 + a \rho}{a \rho (1 - v)^{\rho^{1 / b}}} \right] dv\end{array} [REMOVE_ME_2]$

for $x > 0$, $0 < p < 1$, $a > 0$, the first scale parameter, $b > 0$, the second scale parameter, and $\rho > 0$, the shape parameter.

Description

Computes the pdf, cdf, value at risk and expected shortfall for the Beard distribution due to Beard (1959) given by

for $x > 0$, $0 < p < 1$, $a > 0$, the first scale parameter, $b > 0$, the second scale parameter, and $\rho > 0$, the shape parameter.

dbeard(x, a=1, b=1, rho=1, log=FALSE)
pbeard(x, a=1, b=1, rho=1, log.p=FALSE, lower.tail=TRUE)
varbeard(p, a=1, b=1, rho=1, log.p=FALSE, lower.tail=TRUE)
esbeard(p, a=1, b=1, rho=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
• a: the value of the first scale parameter, must be positive, the default is 1
• b: the value of the second scale parameter, must be positive, the default is 1
• rho: 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)
dbeard(x)
pbeard(x)
varbeard(x)
esbeard(x)