Computes the pdf, cdf, value at risk and expected shortfall for the Hosking logistic distribution due to Hosking (1989, 1990) given by

*\begin{array}{ll}
&\displaystyle
f (x) = \frac {(1 - k x)^{1 / k - 1}}{≤ft[ 1 + (1 - k x)^{1 / k} \right]^2},
\\
&\displaystyle
F (x) = \frac {1}{1 + (1 - k x)^{1 / k}},
\\
&\displaystyle
{\rm VaR}_p (X) = \frac {1}{k} ≤ft[ 1 - ≤ft( \frac {1 - p}{p} \right)^k \right],
\\
&\displaystyle
{\rm ES}_p (X) = \frac {1}{k} - \frac {1}{kp} B_p (1 - k, 1 + k)
\end{array}*

for *x < 1/k* if *k > 0*, *x > 1/k* if *k < 0*, *-∞ < x < ∞* if *k = 0*,
and *-∞ < k < ∞*, the shape 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 |

`k` |
the value of the 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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