Burr distribution

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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}} ≤ft[ 1 + ≤ft( x / a \right)^{-b} \right]^{-2}, \\ &\displaystyle F (x) = \frac {1}{1 + ≤ft( 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 ≤ft( 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

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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

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)

Saralees Nadarajah

References

S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall, submitted

Examples

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

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