Description Usage Arguments Details Value Note Author(s) References See Also
GEV
provides the link between L-moments of a sample and the three parameter
generalized extreme value distribution.
1 2 3 4 5 6 | 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)
|
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 |
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
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.
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.
Alberto Viglione, e-mail: alviglio@tiscali.it.
Hosking, J.R.M. and Wallis, J.R. (1997) Regional Frequency Analysis: an approach based on L-moments, Cambridge University Press, Cambridge, UK.
rnorm
, runif
, KAPPA
, Lmoments
.
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