Generalized Pareto {POT} | R Documentation |
The Generalized Pareto Distribution
Description
Density, distribution function, quantile function and random generation for the GP distribution with location equal to 'loc', scale equal to 'scale' and shape equal to 'shape'.
Usage
rgpd(n, loc = 0, scale = 1, shape = 0)
pgpd(q, loc = 0, scale = 1, shape = 0, lower.tail = TRUE, lambda = 0)
qgpd(p, loc = 0, scale = 1, shape = 0, lower.tail = TRUE, lambda = 0)
dgpd(x, loc = 0, scale = 1, shape = 0, log = FALSE)
Arguments
x , q |
vector of quantiles. |
p |
vector of probabilities. |
n |
number of observations. |
loc |
vector of the location parameters. |
scale |
vector of the scale parameters. |
shape |
a numeric of the shape parameter. |
lower.tail |
logical; if TRUE (default), probabilities are |
log |
logical; if TRUE, probabilities p are given as log(p). |
lambda |
a single probability - see the "value" section. |
Value
If 'loc', 'scale' and 'shape' are not specified they assume the default values of '0', '1' and '0', respectively.
The GP distribution function for loc = u
, scale = \sigma
and shape = \xi
is
G(x) = 1 - \left[ 1 + \frac{\xi (x - u )}{ \sigma } \right] ^ { - 1 /
\xi}
for 1 + \xi ( x - u ) / \sigma > 0
and x >
u
, where \sigma > 0
. If \xi = 0
, the distribution is defined by continuity corresponding to the
exponential distribution.
By definition, the GP distribution models exceedances above a
threshold. In particular, the G
function is a suited
candidate to model
\Pr\left[ X \geq x | X > u \right] = 1 - G(x)
for u
large enough.
However, it may be usefull to model the "non conditional" quantiles,
that is the ones related to \Pr[ X \leq x]
. Using
the conditional probability definition, one have :
\Pr\left[ X \geq x \right] = \left(1 - \lambda\right) \left( 1 +
\xi \frac{x - u}{\sigma}\right)^{-1/\xi}
where \lambda = \Pr[ X \leq u]
.
When \lambda = 0
, the "conditional" distribution
is equivalent to the "non conditional" distribution.
Examples
dgpd(0.1)
rgpd(100, 1, 2, 0.2)
qgpd(seq(0.1, 0.9, 0.1), 1, 0.5, -0.2)
pgpd(12.6, 2, 0.5, 0.1)
##for non conditional quantiles
qgpd(seq(0.9, 0.99, 0.01), 1, 0.5, -0.2, lambda = 0.9)
pgpd(2.6, 2, 2.5, 0.25, lambda = 0.5)