TL2 {VaRES}

Topp-Leone distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the Topp-Leone distribution due to Topp and Leone (1955) given by

\begin{array}{ll} &\displaystyle f(x) = 2 b (x (2 - x))^{b - 1} (1 - x), \\ &\displaystyle F(x) = (x (2 - x))^b, \\ &\displaystyle {\rm VaR}_p (X) = 1 - \sqrt{1 - p^{1 / b}}, \\ &\displaystyle {\rm ES}_p (X) = 1 - \frac {b}{p} B_{p^{1 / b}} \left( b, \frac {3}{2} \right) \end{array}

for x > 0, 0 < p < 1, and b > 0, the shape parameter.

Usage

dTL2(x, b=1, log=FALSE)
pTL2(x, b=1, log.p=FALSE, lower.tail=TRUE)
varTL2(p, b=1, log.p=FALSE, lower.tail=TRUE)
esTL2(p, 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

b

the value of the 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)
dTL2(x)
pTL2(x)
varTL2(x)
esTL2(x)