MRbeta function

McDonald-Richards beta distribution

Computes the pdf, cdf, value at risk and expected shortfall for the McDonald-Richards beta distribution due to McDonald and Richards (1987a, 1987b) given by [REMOVE_ME]$\begin{array}{ll}&\displaystylef (x) = \frac {x^{ar - 1} \left( bq^r - x^r \right)^{b - 1}}{\left( b q^r \right)^{a + b - 1} B (a, b)},\\&\displaystyleF (x) = I_{\frac {x^r}{b q^r}} (a, b),\\&\displaystyle{\rm VaR}_p (X) = b^{1/r} q \left[ I_p^{-1} (a, b) \right]^{1/r},\\&\displaystyle{\rm ES}_p (X) = \frac {b^{1/r} q}{p} \int_0^p \left[ I_v^{-1} (a, b) \right]^{1/r} dv\end{array} [REMOVE_ME_2]$

for $0 \leq x \leq b^{1 / r} q$, $0 < p < 1$, $q > 0$, the scale parameter, $a > 0$, the first shape parameter, $b > 0$, the second shape parameter, and $r > 0$, the third shape parameter.

Description

Computes the pdf, cdf, value at risk and expected shortfall for the McDonald-Richards beta distribution due to McDonald and Richards (1987a, 1987b) given by

for $0 \leq x \leq b^{1 / r} q$, $0 < p < 1$, $q > 0$, the scale parameter, $a > 0$, the first shape parameter, $b > 0$, the second shape parameter, and $r > 0$, the third shape parameter.

dMRbeta(x, a=1, b=1, r=1, q=1, log=FALSE)
pMRbeta(x, a=1, b=1, r=1, q=1, log.p=FALSE, lower.tail=TRUE)
varMRbeta(p, a=1, b=1, r=1, q=1, log.p=FALSE, lower.tail=TRUE)
esMRbeta(p, a=1, b=1, r=1, q=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
• q: 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
• b: the value of the second shape parameter, must be positive, the default is 1
• r: 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

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)
dMRbeta(x)
pMRbeta(x)
varMRbeta(x)
esMRbeta(x)