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 0.6-9 Index]