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

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

for *c ≤q x ≤q d*, *0 < p < 1*, *a > 0*, the first shape parameter, *b > 0*, the second shape parameter, *-∞ < c < ∞*, the first location parameter, and *-∞ < c < d < ∞*, the second location parameter.

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`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 |

`c` |
the value of the first location parameter, can take any real value, the default is zero |

`d` |
the value of the second location parameter, can take any real value but must be greater than c, 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 |

`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 |

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`

.

Saralees Nadarajah

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

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