loglogis function

Log-logistic distribution

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

for $x > 0$, $0 < p < 1$, $a > 0$, the scale parameter, and $b > 0$, the shape parameter, where $B_x (a, b) = \int_0^x t^{a - 1} (1 - t)^{b - 1} dt$ denotes the incomplete beta function.

Description

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

for $x > 0$, $0 < p < 1$, $a > 0$, the scale parameter, and $b > 0$, the shape parameter, where $B_x (a, b) = \int_0^x t^{a - 1} (1 - t)^{b - 1} dt$ denotes the incomplete beta function.

dloglogis(x, a=1, b=1, log=FALSE)
ploglogis(x, a=1, b=1, log.p=FALSE, lower.tail=TRUE)
varloglogis(p, a=1, b=1, log.p=FALSE, lower.tail=TRUE)
esloglogis(p, 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
• a: the value of the scale parameter, must be positive, the default is 1
• 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

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)
dloglogis(x)
ploglogis(x)
varloglogis(x)
esloglogis(x)