Description Usage Arguments Value Author(s) References Examples

Computes the pdf, cdf, value at risk and expected shortfall for the Kumaraswamy distribution due to Kumaraswamy (1980) given by

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

for *0 < x < 1*, *0 < p < 1*, *a > 0*, the first shape parameter, and *b > 0*, the second 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 |

`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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