GENPAR {nsRFA}R Documentation

Three parameter generalized Pareto distribution and L-moments

Description

GENPAR provides the link between L-moments of a sample and the three parameter generalized Pareto distribution.

Usage

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

Arguments

x

vector of quantiles

xi

vector of genpar location parameters

alfa

vector of genpar scale parameters

k

vector of genpar 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/Pareto_distribution for an introduction to the Pareto distribution.

Definition

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

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

Probability density function:

f(x) = α^{-1} e^{-(1-k)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) = 1-e^{-y}

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

k=0 is the exponential distribution; k=1 is the uniform distribution on the interval ξ < x ≤ ξ + α.

L-moments

L-moments are defined for k>-1.

λ_1 = ξ + α/(1+k)]

λ_2 = α/[(1+k)(2+k)]

τ_3 = (1-k)/(3+k)

τ_4 = (1-k)(2-k)/[(3+k)(4+k)]

The relation between τ_3 and τ_4 is given by

τ_4 = \frac{τ_3 (1 + 5 τ_3)}{5+τ_3}

Parameters

If ξ is known, k=(λ_1 - ξ)/λ_2 - 2 and α=(1+k)(λ_1 - ξ); if ξ is unknown, k=(1 - 3 τ_3)/(1 + τ_3), α=(1+k)(2+k)λ_2 and ξ=λ_1 - (2+k)λ_2.

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

Value

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

Examples

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

camp <- split(x,fac)$"45"
ll <- Lmoments(camp)
parameters <- par.genpar(ll[1],ll[2],ll[4])
f.genpar(1800,parameters$xi,parameters$alfa,parameters$k)
F.genpar(1800,parameters$xi,parameters$alfa,parameters$k)
invF.genpar(0.7161775,parameters$xi,parameters$alfa,parameters$k)
Lmom.genpar(parameters$xi,parameters$alfa,parameters$k)
rand.genpar(100,parameters$xi,parameters$alfa,parameters$k)

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

[Package nsRFA version 0.7-14 Index]