| MRbeta {VaRES} | R Documentation |
McDonald-Richards beta distribution
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
\begin{array}{ll}
&\displaystyle
f (x) = \frac {x^{ar - 1} \left( bq^r - x^r \right)^{b - 1}}
{\left( b q^r \right)^{a + b - 1} B (a, b)},
\\
&\displaystyle
F (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}
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.
Usage
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 |
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)
Saralees Nadarajah
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, doi:10.1080/03610918.2014.944658
Examples
x=runif(10,min=0,max=1)
dMRbeta(x)
pMRbeta(x)
varMRbeta(x)
esMRbeta(x)