# Beta Burr XII distribution

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### Description

Computes the pdf, cdf, value at risk and expected shortfall for the beta Burr XII distribution given by

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

for x > 0, 0 < p < 1, a > 0, the first shape parameter, b > 0, the second shape parameter, c > 0, the third shape parameter, and k > 0, the fourth shape parameter.

### Usage

 1 2 3 4 dbetaburr7(x, a=1, b=1, c=1, k=1, log=FALSE) pbetaburr7(x, a=1, b=1, c=1, k=1, log.p=FALSE, lower.tail=TRUE) varbetaburr7(p, a=1, b=1, c=1, k=1, log.p=FALSE, lower.tail=TRUE) esbetaburr7(p, a=1, b=1, c=1, k=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 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 c the value of the third shape parameter, must be positive, the default is 1 k the value of the fourth 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)

 1 2 3 4 5 x=runif(10,min=0,max=1) dbetaburr7(x) pbetaburr7(x) varbetaburr7(x) esbetaburr7(x)