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 {λ \exp (-b λ x)}{B (a, b)} ≤ft[ 1 - \exp (-λ x) \right]^{a - 1}, \\ &\displaystyle F (x) = I_{1 - \exp (-λ x)} (a, b), \\ &\displaystyle {\rm VaR}_p (X) = -\frac {1}{λ} \log ≤ft[ 1 - I_p^{-1} (a, b) \right], \\ &\displaystyle {\rm ES}_p (X) = -\frac {1}{p λ} \int_0^p \log ≤ft[ 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 λ > 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

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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

S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall, submitted

Examples

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