GPD {qrmtools} | R Documentation |
Density, distribution function, quantile function and random variate generation for the (generalized) Pareto distribution (GPD).
dGPD(x, shape, scale, log = FALSE) pGPD(q, shape, scale, lower.tail = TRUE, log.p = FALSE) qGPD(p, shape, scale, lower.tail = TRUE, log.p = FALSE) rGPD(n, shape, scale) dPar(x, shape, scale = 1, log = FALSE) pPar(q, shape, scale = 1, lower.tail = TRUE, log.p = FALSE) qPar(p, shape, scale = 1, lower.tail = TRUE, log.p = FALSE) rPar(n, shape, scale = 1)
x, q |
vector of quantiles. |
p |
vector of probabilities. |
n |
number of observations. |
shape |
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 |
|
log, log.p |
logical; if |
The distribution function of the generalized Pareto distribution is given by
F(x) = 1-(1+xi x/beta)^{-1/xi} if xi != 0 and 1-exp(-x/beta) if xi = 0,
where beta>0 and x >= 0 if xi >= 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 >= 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.
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.
Marius Hofert
McNeil, A. J., Frey, R., and Embrechts, P. (2015). Quantitative Risk Management: Concepts, Techniques, Tools. Princeton University Press.
## Basic sanity checks curve(dGPD(x, shape = 0.5, scale = 3), from = -1, to = 5) plot(pGPD(rGPD(1000, shape = 0.5, scale = 3), shape = 0.5, scale = 3)) # should be U[0,1]