GENLOGIS {nsRFA}

Three parameter generalized logistic distribution and L-moments

Description

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

Usage

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

Arguments

x

vector of quantiles

xi

vector of genlogis location parameters

alfa

vector of genlogis scale parameters

k

vector of genlogis 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/Logistic_distribution for an introduction to the Logistic Distribution.

Definition

Parameters (3): \xi (location), \alpha (scale), k (shape).

Range of x: -\infty < x \le \xi + \alpha / k if k>0; -\infty < x < \infty if k=0; \xi + \alpha / k \le x < \infty if k<0.

Probability density function:

f(x) = \frac{\alpha^{-1} e^{-(1-k)y}}{(1+e^{-y})^2}

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/(1+e^{-y})

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

k=0 is the logistic distribution.

L-moments

L-moments are defined for -1<k<1.

\lambda_1 = \xi + \alpha[1/k - \pi / \sin (k \pi)]

\lambda_2 = \alpha k \pi / \sin (k \pi)

\tau_3 = -k

\tau_4 = (1+5 k^2)/6

Parameters

k=-\tau_3, \alpha = \frac{\lambda_2 \sin (k \pi)}{k \pi}, \xi = \lambda_1 - \alpha (\frac{1}{k} - \frac{\pi}{\sin (k \pi)}).

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

Value

f.genlogis gives the density f, F.genlogis gives the distribution function F, invF.genlogis gives the quantile function x, Lmom.genlogis gives the L-moments (\lambda_1, \lambda_2, \tau_3, \tau_4), par.genlogis gives the parameters (xi, alfa, k), and rand.genlogis generates random deviates.

Note

For information on the package and the Author, and for all the references, see nsRFA.

See Also

rnorm, runif, EXP, GENPAR, 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)$"45"
ll <- Lmoments(camp)
parameters <- par.genlogis(ll[1],ll[2],ll[4])
f.genlogis(1800,parameters$xi,parameters$alfa,parameters$k)
F.genlogis(1800,parameters$xi,parameters$alfa,parameters$k)
invF.genlogis(0.7697433,parameters$xi,parameters$alfa,parameters$k)
Lmom.genlogis(parameters$xi,parameters$alfa,parameters$k)
rand.genlogis(100,parameters$xi,parameters$alfa,parameters$k)

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