GPD shape parameter \(\xi\) (a real number) and
Pareto shape parameter \(\theta\) (a positive number).
scale
GPD scale parameter \(\beta\) (a positive number)
and Pareto scale parameter \(\kappa\) (a positive number).
lower.tail
logical; if TRUE (default)
probabilities are \(P(X \le x)\) otherwise, \(P(X > x)\).
log, log.p
logical; if TRUE, probabilities p are
given as log(p).
Value
dGPD() computes the density, pGPD() the distribution
function, qGPD() the quantile function and rGPD() random
variates of the generalized Pareto distribution.
Similary for dPar(), pPar(), qPar() and
rPar() for the Pareto distribution.
Details
The distribution function of the generalized Pareto distribution is given by
$$F(x) = \cases{
1-(1+\xi x/\beta)^{-1/\xi},&if $\xi\neq 0$,\cr
1-\exp(-x/\beta),&if $\xi = 0$,\cr}$$
where \(\beta>0\) and \(x\ge0\) if \(\xi\ge
0\)
and \(x\in[0,-\beta/\xi]\) if \(\xi<0\).
The distribution function of the Pareto distribution is given by
$$F(x) = 1-(1+x/\kappa)^{-\theta},\ x\ge 0,$$ where \(\theta > 0\), \(\kappa > 0\).
In contrast to dGPD(), pGPD(), qGPD() and
rGPD(), the functions dPar(), pPar(),
qPar() and rPar() are vectorized in their main
argument and the parameters.
References
McNeil, A. J., Frey, R., and Embrechts, P. (2015).
Quantitative Risk Management: Concepts, Techniques, Tools.
Princeton University Press.