Computes the pdf, cdf, value at risk and expected shortfall for the Freimer distribution due to Freimer et al. (1988) given by

*\begin{array}{ll}
&\displaystyle
{\rm VaR}_p (X) = \frac {1}{a} ≤ft[ \frac {p^b - 1}{b} -
\frac {(1 - p)^c - 1}{c} \right],
\\
&\displaystyle
{\rm ES}_p (X) = \frac {1}{a} ≤ft( \frac {1}{c} - \frac {1}{b} \right) +
\frac {p^b}{a b (b + 1)} + \frac {(1 - p)^{c + 1} - 1}{p a c (c + 1)}
\end{array}*

for *0 < p < 1*, *a > 0*, the scale parameter,
*b > 0*, the first shape parameter, and *c > 0*, the second shape parameter.

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`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 |

`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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