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]\displaystylef(x)=B(a,b)λexp(−bλx)[1−exp(−λx)]a−1,\displaystyleF(x)=I1−exp(−λx)(a,b),VaRp(X)=−λ1log[1−Ip−1(a,b)],ESp(X)=−pλ1∫0plog[1−Iv−1(a,b)]dv[REMOVEME2]
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 Ix(a,b)=∫0xta−1(1−t)b−1dt/B(a,b) denotes the incomplete beta function ratio, B(a,b)=∫01ta−1(1−t)b−1dt denotes the beta function, and Ix−1(a,b) denotes the inverse function of Ix(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
\displaystylef(x)=B(a,b)λexp(−bλx)[1−exp(−λx)]a−1,\displaystyleF(x)=I1−exp(−λx)(a,b),VaRp(X)=−λ1log[1−Ip−1(a,b)],ESp(X)=−pλ1∫0plog[1−Iv−1(a,b)]dv
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 Ix(a,b)=∫0xta−1(1−t)b−1dt/B(a,b) denotes the incomplete beta function ratio, B(a,b)=∫01ta−1(1−t)b−1dt denotes the beta function, and Ix−1(a,b) denotes the inverse function of Ix(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)