betaburr {VaRES}

Beta Burr distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the beta Burr distribution due to Parana\'iba et al. (2011) given by

\begin{array}{ll} &\displaystyle f (x) = \frac {b a^{bd}}{B (c, d)x^{bd + 1}} \left[ 1 + \left( x / a \right)^{-b} \right]^{-c - d}, \\ &\displaystyle F (x) = I_{\frac {1}{1 + \left( x / a \right)^{-b}}} (c, d), \\ &\displaystyle {\rm VaR}_p (X) = a \left[ I_p^{-1} (c, d) \right]^{1 / b} \left[ 1 - I_p^{-1} (c, d) \right]^{-1 / b}, \\ &\displaystyle {\rm ES}_p (X) = \frac {a}{p} \int_0^p \left[ I_v^{-1} (c, d) \right]^{1 / b} \left[ 1 - I_v^{-1} (c, d) \right]^{-1 / b} dv \end{array}

for x > 0, 0 < p < 1, a > 0, the scale parameter, b > 0, the first shape parameter, c > 0, the second shape parameter, and d > 0, the third shape parameter, where I_x (a, b) = \int_0^x t^{a - 1} (1 - t)^{b - 1} dt / B (a, b) denotes the incomplete beta function ratio, B (a, b) = \int_0^1 t^{a - 1} (1 - t)^{b - 1} dt denotes the beta function, and I_x^{-1} (a, b) denotes the inverse function of I_x (a, b).

Usage

dbetaburr(x, a=1, b=1, c=1, d=1, log=FALSE)
pbetaburr(x, a=1, b=1, c=1, d=1, log.p=FALSE, lower.tail=TRUE)
varbetaburr(p, a=1, b=1, c=1, d=1, log.p=FALSE, lower.tail=TRUE)
esbetaburr(p, a=1, b=1, c=1, d=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 first shape parameter, must be positive, the default is 1

c

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

d

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

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, doi:10.1080/03610918.2014.944658

Examples

x=runif(10,min=0,max=1)
dbetaburr(x)
pbetaburr(x)
varbetaburr(x)
esbetaburr(x)