VaRES (version 1.0.2)

burr: Burr distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the Burr distribution due to Burr (1942) given by $$\begin{array}{ll} &\displaystyle f (x) = \frac {b a^b}{x^{b + 1}} \left[ 1 + \left( x / a \right)^{-b} \right]^{-2}, \\ &\displaystyle F (x) = \frac {1}{1 + \left( x / a \right)^{-b}}, \\ &\displaystyle {\rm VaR}_p (X) = a p^{1 / b} (1 - p)^{-1 / b}, \\ &\displaystyle {\rm ES}_p (X) = \frac {a}{p} B_p \left( 1 / b + 1, 1 - 1 / b \right) \end{array}$$ 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

dburr(x, a=1, b=1, log=FALSE)
pburr(x, a=1, b=1, log.p=FALSE, lower.tail=TRUE)
varburr(p, a=1, b=1, log.p=FALSE, lower.tail=TRUE)
esburr(p, a=1, b=1)

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.

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

Author

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, tools:::Rd_expr_doi("10.1080/03610918.2014.944658")

Examples

x=runif(10,min=0,max=1)
dburr(x)
pburr(x)
varburr(x)
esburr(x)