Description Usage Arguments Value Author(s) References Examples

Computes the pdf, cdf, value at risk and expected shortfall for the log-logistic distribution given by

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

for *x > 0*, *0 < p < 1*, *a > 0*, the scale parameter, and *b > 0*, the shape parameter,
where *B_x (a, b) = \int_0^x t^{a - 1} (1 - t)^{b - 1} dt* denotes the incomplete beta function.

1 2 3 4 | ```
dloglogis(x, a=1, b=1, log=FALSE)
ploglogis(x, a=1, b=1, log.p=FALSE, lower.tail=TRUE)
varloglogis(p, a=1, b=1, log.p=FALSE, lower.tail=TRUE)
esloglogis(p, a=1, b=1)
``` |

`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 scale parameter, must be positive, the default is 1 |

`b` |
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

1 2 3 4 5 | ```
x=runif(10,min=0,max=1)
dloglogis(x)
ploglogis(x)
varloglogis(x)
esloglogis(x)
``` |

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