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): xi (location), α (scale), k (shape).

Range of x: xi < x <= xi + α / k if k>0; xi <= x < infty if k <= 0.

Probability density function:

f(x) = α^{-1} e^{-(1-k)y}

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

Cumulative distribution function:

F(x) = 1-e^{-y}

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

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

L-moments

L-moments are defined for k>-1.

λ_1 = xi + α/(1+k)]

λ_2 = α/[(1+k)(2+k)]

tau_3 = (1-k)/(3+k)

tau_4 = (1-k)(2-k)/[(3+k)(4+k)]

The relation between tau_3 and tau_4 is given by

tau_4 = frac{tau_3 (1 + 5 tau_3)}{5+tau_3}

Parameters

If xi is known, k=(λ_1 - xi)/λ_2 - 2 and α=(1+k)(λ_1 - xi); if xi is unknown, k=(1 - 3 tau_3)/(1 + tau_3), α=(1+k)(2+k)λ_2 and xi=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, tau_3, tau_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 0.6-7 Index]