VaRES (version 1.0.2)

betaburr: 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 $$\begin{array}{ll}& {\displaystyle f(x)=\frac{b{a}^{bd}}{B(c,d)x^{bd+1}}{[1+{\left(x/a\right)}^{-b}]}^{-c-d},}\\ & {\displaystyle F(x)=I_{\frac{1}{1+{\left(x/a\right)}^{-b}}}(c,d),}\\ & {\displaystyle {\mathrm{V}\mathrm{a}\mathrm{R}}_p(X)=a{[I_p^{-1}(c,d)]}^{1/b}{[1-I_p^{-1}(c,d)]}^{-1/b},}\\ & {\displaystyle {\mathrm{E}\mathrm{S}}_p(X)=\frac{a}{p}{\int }_0^p{[I_v^{-1}(c,d)]}^{1/b}{[1-I_v^{-1}(c,d)]}^{-1/b}dv}\end{array}$$ 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)

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 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

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)
dbetaburr(x)
pbetaburr(x)
varbetaburr(x)
esbetaburr(x)