logisrayleigh function

Logistic Rayleigh distribution

Computes the pdf, cdf, value at risk and expected shortfall for the logistic Rayleigh distribution due to Lan and Leemis (2008) given by [REMOVE_ME]$\begin{array}{ll}&\displaystylef(x) = a \lambda x \exp \left( \lambda x^2 / 2 \right)\left[ \exp \left( \lambda x^2 / 2 \right) - 1 \right]^{a - 1}\left\{ 1 + \left[ \exp \left( \lambda x^2 / 2 \right) - 1 \right]^a \right\}^{-2},\\&\displaystyleF(x) = \frac {\left[ \exp \left( \lambda x^2 / 2 \right) - 1 \right]^a}{1 + \left[ \exp \left( \lambda x^2 / 2 \right) - 1 \right]^a},\\&\displaystyle{\rm VaR}_p (X) = \sqrt{\frac {2}{\lambda}}\sqrt{\log \left[ 1 + \left( \frac {p}{1 - p} \right)^{1 / a} \right]},\\&\displaystyle{\rm ES}_p (X) = \frac {\sqrt{2}}{p \sqrt{\lambda}}\int_0^p \left\{ \log \left[ 1 + \left( \frac {v}{1 - v} \right)^{1 / a} \right] \right\}^{1/2} dv\end{array} [REMOVE_ME_2]$

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

Description

Computes the pdf, cdf, value at risk and expected shortfall for the logistic Rayleigh distribution due to Lan and Leemis (2008) given by

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

dlogisrayleigh(x, a=1, lambda=1, log=FALSE)
plogisrayleigh(x, a=1, lambda=1, log.p=FALSE, lower.tail=TRUE)
varlogisrayleigh(p, a=1, lambda=1, log.p=FALSE, lower.tail=TRUE)
eslogisrayleigh(p, a=1, lambda=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 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)
dlogisrayleigh(x)
plogisrayleigh(x)
varlogisrayleigh(x)
eslogisrayleigh(x)