# Talk:Generalized logistic distribution

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## Images for this distribution

I've made a start, tho I'm not yet happy with the results...

# R source code for Wikipedia SVG plot of Skew Logistic distribution, CDF
# by User:Tayste, February 2010.  Public Domain.

graphics.off()
require("RSvgDevice")

W = 8; H = 6
if (!interactive()) devSVG("SkewLogisticCDF.svg",width=W,height=H)
else windows(width=W, height=H)

x = seq(-10, 10, by=0.1)
alpha = c(0.2, 0.5, 1, 2, 5)
colour <- hcl(seq(0,360,length=1+length(alpha))[-1],l=50)
legend.text = character(0)

par(mar=c(3.3,3,0.5,0.5))
plot(1, xlim=range(x), ylim=c(0,1), type="n", xlab="", ylab="", axes=FALSE)
axis(1)
axis(2, las=1)
abline(v=0,col='grey80')
abline(h=0,col='grey80')

for (i in seq_along(alpha)) {
cdf <- (1 + exp(-x)) ^ (-alpha[i])
lines(x, cdf, col=colour[i], lty=i, lwd=2)
}

legend(-10, 0.8, lty=1:5, lwd=2, col=colour, bty='n',
c(expression(alpha == 0.2),
expression(alpha == 0.5),
expression(alpha == 1),
expression(alpha == 2),
expression(alpha == 5)
)
)

if (!interactive()) dev.off()



Tayste (edits) 21:25, 9 February 2010 (UTC)

## More variants?

The "lmom" R package seems to use yet another variant:

$F(x) = \frac{1}{1+e^{-y}}$ (i.e. logistic distribution cdf in $y$)

where

$y=-k^{-1} \log \left(1-k\cdot \frac{x-\mu}{s}\right)$

Apparently, this is a special case of what is called "kappa distribution" there.

See "lmomco" package documentation for details.

Substituting $y$ and simplifying the formula yields:

$F(x) = \frac{1}{1+\left(1-k\cdot \frac{x-\mu}{s}\right)^{-k}}$

Dropping location and scale:

$F(x; \alpha) = \frac{1}{1+\left(1-\alpha\cdot x\right)^{-\alpha}}$

--Chire (talk) 18:37, 26 August 2013 (UTC)