| 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 https://en.wikipedia.org/wiki/Pareto_distribution for an introduction to the Pareto distribution.
Definition
Parameters (3): \xi (location), \alpha (scale), k (shape).
Range of x: \xi < x \le \xi + \alpha / k if k>0;
\xi \le x < \infty if k \le 0.
Probability density function:
f(x) = \alpha^{-1} e^{-(1-k)y}
where y = -k^{-1}\log\{1 - k(x - \xi)/\alpha\} if k \ne 0,
y = (x-\xi)/\alpha if k=0.
Cumulative distribution function:
F(x) = 1-e^{-y}
Quantile function:
x(F) = \xi + \alpha[1-(1-F)^k]/k if k \ne 0,
x(F) = \xi - \alpha \log(1-F) if k=0.
k=0 is the exponential distribution; k=1 is the uniform distribution on the interval \xi < x \le \xi + \alpha.
L-moments
L-moments are defined for k>-1.
\lambda_1 = \xi + \alpha/(1+k)]
\lambda_2 = \alpha/[(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=(\lambda_1 - \xi)/\lambda_2 - 2 and \alpha=(1+k)(\lambda_1 - \xi);
if \xi is unknown, k=(1 - 3 \tau_3)/(1 + \tau_3), \alpha=(1+k)(2+k)\lambda_2 and
\xi=\lambda_1 - (2+k)\lambda_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 (\lambda_1, \lambda_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)