GUMBEL {nsRFA}R Documentation

Two parameter Gumbel distribution and L-moments

Description

GUMBEL provides the link between L-moments of a sample and the two parameter Gumbel distribution.

Usage

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)

Arguments

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

Details

See http://en.wikipedia.org/wiki/Fisher-Tippett_distribution for an introduction to the Gumbel distribution.

Definition

Parameters (2): xi (location), α (scale).

Range of x: -infty < x < infty.

Probability density function:

f(x) = α^{-1} exp[-(x-xi)/α] exp{- exp[-(x-xi)/α]}

Cumulative distribution function:

F(x) = exp[-exp(-(x-xi)/α)]

Quantile function: x(F) = xi - α log(-log F).

L-moments

λ_1 = xi + α gamma

λ_2 = α 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

α=λ_2 / log 2

xi = λ_1 - gamma α

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.

Value

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 (λ_1, λ_2, tau_3, tau_4)), par.gumb gives the parameters (xi, alfa), and rand.gumb generates random deviates.

Note

For information on the package and the Author, and for all the references, see nsRFA.

See Also

rnorm, runif, EXP, GENLOGIS, GENPAR, GEV, KAPPA, LOGNORM, P3; DISTPLOTS, GOFmontecarlo, Lmoments.

Examples

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)

[Package nsRFA version 0.6-7 Index]