# loglogis: Log-logistic distribution In VaRES: Computes value at risk and expected shortfall for over 100 parametric distributions

## Description

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

\begin{array}{ll} &\displaystyle f (x) = \frac {b a^b x^{b - 1}} {≤ft( a^b + x^b \right)^2}, \\ &\displaystyle F (x) = \frac {x^b}{a^b + x^b}, \\ &\displaystyle {\rm VaR}_p (X) = a ≤ft( \frac {p}{1 - p} \right)^{1 / b}, \\ &\displaystyle {\rm ES}_p (X) = \frac {a}{p} B_p ≤ft( 1 + \frac {1}{b}, 1 - \frac {1}{b} \right) \end{array}

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.

## Usage

 1 2 3 4 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

## 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)

 1 2 3 4 5 x=runif(10,min=0,max=1) dloglogis(x) ploglogis(x) varloglogis(x) esloglogis(x)