GENLOGIS: 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

vector of quantiles

vector of genlogis location parameters

alfa

vector of genlogis scale parameters

vector of genlogis shape parameters

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

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.

Details

See http://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.

Examples

Run this code

# NOT RUN {
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)
# }

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