GUMBEL | R Documentation |
GUMBEL
provides the link between L-moments of a sample and the two parameter
Gumbel distribution.
f.gumb (x, xi, alfa)
F.gumb (x, xi, alfa)
invF.gumb (F, xi, alfa)
Lmom.gumb (xi, alfa)
par.gumb (lambda1, lambda2)
rand.gumb (numerosita, xi, alfa)
x |
vector of quantiles |
xi |
vector of gumb location parameters |
alfa |
vector of gumb scale parameters |
F |
vector of probabilities |
lambda1 |
vector of sample means |
lambda2 |
vector of L-variances |
numerosita |
numeric value indicating the length of the vector to be generated |
See https://en.wikipedia.org/wiki/Fisher-Tippett_distribution for an introduction to the Gumbel distribution.
Definition
Parameters (2): \xi
(location), \alpha
(scale).
Range of x
: -\infty < x < \infty
.
Probability density function:
f(x) = \alpha^{-1} \exp[-(x-\xi)/\alpha] \exp\{- \exp[-(x-\xi)/\alpha]\}
Cumulative distribution function:
F(x) = \exp[-\exp(-(x-\xi)/\alpha)]
Quantile function:
x(F) = \xi - \alpha \log(-\log F)
.
L-moments
\lambda_1 = \xi + \alpha \gamma
\lambda_2 = \alpha \log 2
\tau_3 = 0.1699 = \log(9/8)/ \log 2
\tau_4 = 0.1504 = (16 \log 2 - 10 \log 3)/ \log 2
Here \gamma
is Euler's constant, 0.5772...
Parameters
\alpha=\lambda_2 / \log 2
\xi = \lambda_1 - \gamma \alpha
Lmom.gumb
and par.gumb
accept input as vectors of equal length. In f.gumb
, F.gumb
, invF.gumb
and rand.gumb
parameters (xi
, alfa
) must be atomic.
f.gumb
gives the density f
, F.gumb
gives the distribution function F
, invF.gumb
gives
the quantile function x
, Lmom.gumb
gives the L-moments (\lambda_1
, \lambda_2
, \tau_3
, \tau_4
)), par.gumb
gives the parameters (xi
, alfa
), and rand.gumb
generates random deviates.
For information on the package and the Author, and for all the references, see nsRFA
.
rnorm
, runif
, EXP
, GENLOGIS
, GENPAR
, GEV
, KAPPA
, LOGNORM
, P3
; DISTPLOTS
, GOFmontecarlo
, Lmoments
.
data(hydroSIMN)
annualflows[1:10,]
summary(annualflows)
x <- annualflows["dato"][,]
fac <- factor(annualflows["cod"][,])
split(x,fac)
camp <- split(x,fac)$"45"
ll <- Lmoments(camp)
parameters <- par.gumb(ll[1],ll[2])
f.gumb(1800,parameters$xi,parameters$alfa)
F.gumb(1800,parameters$xi,parameters$alfa)
invF.gumb(0.7686843,parameters$xi,parameters$alfa)
Lmom.gumb(parameters$xi,parameters$alfa)
rand.gumb(100,parameters$xi,parameters$alfa)
Rll <- regionalLmoments(x,fac); Rll
parameters <- par.gumb(Rll[1],Rll[2])
Lmom.gumb(parameters$xi,parameters$alfa)
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