Three parameter generalized extreme value distribution and L-moments

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Description

GEV provides the link between L-moments of a sample and the three parameter generalized extreme value distribution.

Usage

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f.GEV (x, xi, alfa, k)
F.GEV (x, xi, alfa, k)
invF.GEV (F, xi, alfa, k)
Lmom.GEV (xi, alfa, k)
par.GEV (lambda1, lambda2, tau3)
rand.GEV (numerosita, xi, alfa, k)

Arguments

x

vector of quantiles

xi

vector of GEV location parameters

alfa

vector of GEV scale parameters

k

vector of GEV shape parameters

F

vector of probabilities

lambda1

vector of sample means

lambda2

vector of L-variances

tau3

vector of L-CA (or L-skewness)

numerosita

numeric value indicating the length of the vector to be generated

Details

See http://en.wikipedia.org/wiki/Generalized_extreme_value_distribution for an introduction to the GEV distribution.

Definition

Parameters (3): ξ (location), α (scale), k (shape).

Range of x: -∞ < x ≤ ξ + α / k if k>0; -∞ < x < ∞ if k=0; ξ + α / k ≤ x < ∞ if k<0.

Probability density function:

f(x) = α^{-1} e^{-(1-k)y - e^{-y}}

where y = -k^{-1}\log\{1 - k(x - ξ)/α\} if k \ne 0, y = (x-ξ)/α if k=0.

Cumulative distribution function:

F(x) = e^{-e^{-y}}

Quantile function: x(F) = ξ + α[1-(-\log F)^k]/k if k \ne 0, x(F) = ξ - α \log(-\log F) if k=0.

k=0 is the Gumbel distribution; k=1 is the reverse exponential distribution.

L-moments

L-moments are defined for k>-1.

λ_1 = ξ + α[1 - Γ (1+k)]/k

λ_2 = α (1-2^{-k}) Γ (1+k)]/k

τ_3 = 2(1-3^{-k})/(1-2^{-k})-3

τ_4 = [5(1-4^{-k})-10(1-3^{-k})+6(1-2^{-k})]/(1-2^{-k})

Here Γ denote the gamma function

Γ (x) = \int_0^{∞} t^{x-1} e^{-t} dt

Parameters

To estimate k, no explicit solution is possible, but the following approximation has accurancy better than 9 \times 10^{-4} for -0.5 ≤ τ_3 ≤ 0.5:

k \approx 7.8590 c + 2.9554 c^2

where

c = \frac{2}{3+τ_3} - \frac{\log 2}{\log 3}

The other parameters are then given by

α = \frac{λ_2 k}{(1-2^{-k})Γ(1+k)}

ξ = λ_1 - α[1 - Γ(1+k)]/k

Lmom.GEV and par.GEV accept input as vectors of equal length. In f.GEV, F.GEV, invF.GEV and rand.GEV parameters (xi, alfa, k) must be atomic.

Value

f.GEV gives the density f, F.GEV gives the distribution function F, invF.GEV gives the quantile function x, Lmom.GEV gives the L-moments (λ_1, λ_2, τ_3, τ_4), par.GEV gives the parameters (xi, alfa, k), and rand.GEV 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, GUMBEL, KAPPA, LOGNORM, P3; DISTPLOTS, GOFmontecarlo, Lmoments.

Examples

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data(hydroSIMN)
annualflows
summary(annualflows)
x <- annualflows["dato"][,]
fac <- factor(annualflows["cod"][,])
split(x,fac)

camp <- split(x,fac)$"45"
ll <- Lmoments(camp)
parameters <- par.GEV(ll[1],ll[2],ll[4])
f.GEV(1800,parameters$xi,parameters$alfa,parameters$k)
F.GEV(1800,parameters$xi,parameters$alfa,parameters$k)
invF.GEV(0.7518357,parameters$xi,parameters$alfa,parameters$k)
Lmom.GEV(parameters$xi,parameters$alfa,parameters$k)
rand.GEV(100,parameters$xi,parameters$alfa,parameters$k)

Rll <- regionalLmoments(x,fac); Rll
parameters <- par.GEV(Rll[1],Rll[2],Rll[4])
Lmom.GEV(parameters$xi,parameters$alfa,parameters$k)

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