burr function

Burr distribution

Computes the pdf, cdf, value at risk and expected shortfall for the Burr distribution due to Burr (1942) given by [REMOVE_ME]$\begin{array}{ll}&\displaystylef (x) = \frac {b a^b}{x^{b + 1}} \left[ 1 + \left( x / a \right)^{-b} \right]^{-2},\\&\displaystyleF (x) = \frac {1}{1 + \left( x / a \right)^{-b}},\\&\displaystyle{\rm VaR}_p (X) = a p^{1 / b} (1 - p)^{-1 / b},\\&\displaystyle{\rm ES}_p (X) = \frac {a}{p} B_p \left( 1 / b + 1, 1 - 1 / b \right)\end{array} [REMOVE_ME_2]$

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 Burr distribution due to Burr (1942) 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.

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)

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)
dburr(x)
pburr(x)
varburr(x)
esburr(x)