logistic function

Logistic distribution

Computes the pdf, cdf, value at risk and expected shortfall for the logistic distribution given by [REMOVE_ME]$\begin{array}{ll}&\displaystylef (x) = \frac {1}{\sigma} \exp \left( -\frac {x - \mu}{\sigma} \right)\left[ 1 + \exp \left( -\frac {x - \mu}{\sigma} \right) \right]^{-2},\\&\displaystyleF (x) = \frac {1}{1 + \exp \left( -\frac {x - \mu}{\sigma} \right)},\\&\displaystyle{\rm VaR}_p (X) = \mu + \sigma \log \left[ p (1 - p) \right],\\&\displaystyle{\rm ES}_p (X) = \mu - 2 \sigma + \sigma \log p - \sigma \frac {1 - p}{p} \log (1 - p)\end{array} [REMOVE_ME_2]$

for $-\infty < x < \infty$, $0 < p < 1$, $-\infty < \mu < \infty$, the location parameter, and $\sigma > 0$, the scale parameter.

Description

Computes the pdf, cdf, value at risk and expected shortfall for the logistic distribution given by

for $-\infty < x < \infty$, $0 < p < 1$, $-\infty < \mu < \infty$, the location parameter, and $\sigma > 0$, the scale parameter.

dlogistic(x, mu=0, sigma=1, log=FALSE)
plogistic(x, mu=0, sigma=1, log.p=FALSE, lower.tail=TRUE)
varlogistic(p, mu=0, sigma=1, log.p=FALSE, lower.tail=TRUE)
eslogistic(p, mu=0, sigma=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
• mu: the value of the location parameter, can take any real value, the default is zero
• sigma: the value of the scale 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)
dlogistic(x)
plogistic(x)
varlogistic(x)
eslogistic(x)