GEV {distributions3} | R Documentation |
Create a Generalised Extreme Value (GEV) distribution
Description
The GEV distribution arises from the Extremal Types Theorem, which is rather
like the Central Limit Theorem (see \link{Normal}
) but it relates to
the maximum of i.i.d. random variables rather than to the sum.
If, after a suitable linear rescaling, the distribution of this maximum
tends to a non-degenerate limit as
tends to infinity then this limit
must be a GEV distribution. The requirement that the variables are independent
can be relaxed substantially. Therefore, the GEV distribution is often used
to model the maximum of a large number of random variables.
Usage
GEV(mu = 0, sigma = 1, xi = 0)
Arguments
mu |
The location parameter, written |
sigma |
The scale parameter, written |
xi |
The shape parameter, written |
Details
We recommend reading this documentation on https://alexpghayes.github.io/distributions3/, where the math will render with additional detail and much greater clarity.
In the following, let be a GEV random variable with location
parameter
mu
= , scale parameter
sigma
= and
shape parameter
xi
= .
Support:
for
;
for
;
and
, the set of all real numbers, for
.
Mean: for
;
for
, where
is Euler's constant, approximately equal to 0.57722; undefined otherwise.
Median: for
;
for
.
Variance:
for
;
for
; undefined otherwise.
Probability density function (p.d.f):
If then
for . The p.d.f. is 0 outside the
support.
In the (Gumbel) special case
for in
, the set of all real numbers.
Cumulative distribution function (c.d.f):
If then
for . The c.d.f. is 0 below the
support and 1 above the support.
In the (Gumbel) special case
for in
, the set of all real numbers.
Value
A GEV
object.
See Also
Other continuous distributions:
Beta()
,
Cauchy()
,
ChiSquare()
,
Erlang()
,
Exponential()
,
FisherF()
,
Frechet()
,
GP()
,
Gamma()
,
Gumbel()
,
LogNormal()
,
Logistic()
,
Normal()
,
RevWeibull()
,
StudentsT()
,
Tukey()
,
Uniform()
,
Weibull()
Examples
set.seed(27)
X <- GEV(1, 2, 0.1)
X
random(X, 10)
pdf(X, 0.7)
log_pdf(X, 0.7)
cdf(X, 0.7)
quantile(X, 0.7)
cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 0.7))