McDonald-Richards beta distribution

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Description

Computes the pdf, cdf, value at risk and expected shortfall for the McDonald-Richards beta distribution due to McDonald and Richards (1987a, 1987b) given by

\begin{array}{ll} &\displaystyle f (x) = \frac {x^{ar - 1} ≤ft( bq^r - x^r \right)^{b - 1}} {≤ft( b q^r \right)^{a + b - 1} B (a, b)}, \\ &\displaystyle F (x) = I_{\frac {x^r}{b q^r}} (a, b), \\ &\displaystyle {\rm VaR}_p (X) = b^{1/r} q ≤ft[ I_p^{-1} (a, b) \right]^{1/r}, \\ &\displaystyle {\rm ES}_p (X) = \frac {b^{1/r} q}{p} \int_0^p ≤ft[ I_v^{-1} (a, b) \right]^{1/r} dv \end{array}

for 0 ≤q x ≤q b^{1 / r} q, 0 < p < 1, q > 0, the scale parameter, a > 0, the first shape parameter, b > 0, the second shape parameter, and r > 0, the third shape parameter.

Usage

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dMRbeta(x, a=1, b=1, r=1, q=1, log=FALSE)
pMRbeta(x, a=1, b=1, r=1, q=1, log.p=FALSE, lower.tail=TRUE)
varMRbeta(p, a=1, b=1, r=1, q=1, log.p=FALSE, lower.tail=TRUE)
esMRbeta(p, a=1, b=1, r=1, q=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

q

the value of the scale parameter, must be positive, the default is 1

a

the value of the first shape parameter, must be positive, the default is 1

b

the value of the second shape parameter, must be positive, the default is 1

r

the value of the third 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)
dMRbeta(x)
pMRbeta(x)
varMRbeta(x)
esMRbeta(x)

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