betaexp function

Beta exponential distribution

Computes the pdf, cdf, value at risk and expected shortfall for the beta exponential distribution due to Nadarajah and Kotz (2006) given by [REMOVE_ME]$\begin{array}{ll}&\displaystylef (x) = \frac {\lambda \exp (-b \lambda x)}{B (a, b)}\left[ 1 - \exp (-\lambda x) \right]^{a - 1},\\&\displaystyleF (x) = I_{1 - \exp (-\lambda x)} (a, b),\\&\displaystyle{\rm VaR}_p (X) = -\frac {1}{\lambda} \log \left[ 1 - I_p^{-1} (a, b) \right],\\&\displaystyle{\rm ES}_p (X) = -\frac {1}{p \lambda} \int_0^p \log \left[ 1 - I_v^{-1} (a, b) \right] dv\end{array} [REMOVE_ME_2]$

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 $I_x (a, b)$.

Description

Computes the pdf, cdf, value at risk and expected shortfall for the beta exponential distribution due to Nadarajah and Kotz (2006) given by

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 $I_x (a, b)$.

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

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)
dbetaexp(x)
pbetaexp(x)
varbetaexp(x)
esbetaexp(x)