Talk:Generalized logistic distribution

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


 * 1) R source code for Wikipedia SVG plot of Skew Logistic distribution, CDF
 * 2) 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)