LNbeta function

Libby-Novick beta distribution

Computes the pdf, cdf, value at risk and expected shortfall for the Libby-Novick beta distribution due to Libby and Novick (1982) given by [REMOVE_ME]$\begin{array}{ll}&\displaystylef (x) = \frac {\lambda^a x^{a - 1} (1 - x)^{b - 1}}{B (a, b) \left[ 1 - (1 - \lambda) x \right]^{a + b}},\\&\displaystyleF (x) = I_{\frac {\lambda x}{1 + (\lambda - 1) x}} (a, b),\\&\displaystyle{\rm VaR}_p (X) = \frac {I_p^{-1} (a, b)}{\lambda - (\lambda - 1) I_p^{-1} (a, b)},\\&\displaystyle{\rm ES}_p (X) = \frac {1}{p} \int_0^p\frac {I_v^{-1} (a, b)}{\lambda - (\lambda - 1) I_v^{-1} (a, b)} dv\end{array} [REMOVE_ME_2]$

for $0 < x < 1$, $0 < p < 1$, $\lambda > 0$, the scale parameter, $a > 0$, the first shape parameter, and $b > 0$, the second shape parameter.

Description

Computes the pdf, cdf, value at risk and expected shortfall for the Libby-Novick beta distribution due to Libby and Novick (1982) given by

for $0 < x < 1$, $0 < p < 1$, $\lambda > 0$, the scale parameter, $a > 0$, the first shape parameter, and $b > 0$, the second shape parameter.

dLNbeta(x, lambda=1, a=1, b=1, log=FALSE)
pLNbeta(x, lambda=1, a=1, b=1, log.p=FALSE, lower.tail=TRUE)
varLNbeta(p, lambda=1, a=1, b=1, log.p=FALSE, lower.tail=TRUE)
esLNbeta(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)
dLNbeta(x)
pLNbeta(x)
varLNbeta(x)
esLNbeta(x)