| betaexp {VaRES} | R Documentation |
Beta exponential distribution
Description
Computes the pdf, cdf, value at risk and expected shortfall for the beta exponential distribution due to Nadarajah and Kotz (2006) given by
\begin{array}{ll}
&\displaystyle
f (x) = \frac {\lambda \exp (-b \lambda x)}{B (a, b)}
\left[ 1 - \exp (-\lambda x) \right]^{a - 1},
\\
&\displaystyle
F (x) = I_{1 - \exp (-\lambda x)} (a, b),
\\
&\displaystyle
{\rm VaR}_p (X) = -\frac {1}{\lambda} \log \left[ 1 - I_p^{-1} (a, b) \right],
\\
&\displaystyle
{\rm ES}_p (X) = -\frac {1}{p \lambda} \int_0^p \log \left[ 1 - I_v^{-1} (a, b) \right] dv
\end{array}
for x > 0, 0 < p < 1, a > 0, the first shape parameter, b > 0, the second shape parameter, and \lambda > 0,
the scale parameter, where I_x (a, b) = \int_0^x t^{a - 1} (1 - t)^{b - 1} dt / B (a, b) denotes
the incomplete beta function ratio,
B (a, b) = \int_0^1 t^{a - 1} (1 - t)^{b - 1} dt denotes the beta function, and
I_x^{-1} (a, b) denotes the inverse function of I_x (a, b).
Usage
dbetaexp(x, lambda=1, a=1, b=1, log=FALSE)
pbetaexp(x, lambda=1, a=1, b=1, log.p=FALSE, lower.tail=TRUE)
varbetaexp(p, lambda=1, a=1, b=1, log.p=FALSE, lower.tail=TRUE)
esbetaexp(p, lambda=1, a=1, b=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 |
lambda |
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 |
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)
dbetaexp(x)
pbetaexp(x)
varbetaexp(x)
esbetaexp(x)