GUMBEL: Two parameter Gumbel distribution and L-moments

In homtest: Homogeneity tests for Regional Frequency Analysis

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): ξ (location), α (scale).

Range of x: -∞ < x < ∞.

Probability density function:

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

Cumulative distribution function:

F(x) = \exp[-\exp(-(x-ξ)/α)]

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

L-moments

λ_1 = ξ + α γ

λ_2 = α \log 2

τ_3 = 0.1699 = \log(9/8)/ \log 2

τ_4 = 0.1504 = (16 \log 2 - 10 \log 3)/ \log 2

Here γ is Euler's constant, 0.5772...

Parameters

α=λ_2 / \log 2

ξ = λ_1 - γ α

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, τ_3, τ_4)), par.gumb gives the parameters (xi, alfa), and rand.gumb generates random deviates.

Note

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.

Author(s)

Alberto Viglione, e-mail: [e-mail].

References

Hosking, J.R.M. and Wallis, J.R. (1997) Regional Frequency Analysis: an approach based on L-moments, Cambridge University Press, Cambridge, UK.

See Also

rnorm, runif, GEV, Lmoments.

homtest documentation built on May 2, 2019, 1:45 p.m.