A generalised form of the logistic distribution is described by
Depending on the sign of δ, the distribution is either lower‐bounded or upper‐bounded.
The CDF is
The CDF can be inverted
Figure 22.1 shows the effect of changing δ with α fixed at 0 and β at 1. Note that all the curves pass through the point (α, 0.25β). Fitting Equation (22.4) to the C4 in propane data gives α as 2.30, β as 1.27 and δ as 0.360. RSS is 0.0529, m\aking the distribution a relative poor choice.
The mean and variance are
As δ → 0, from Equation (11.4), we obtain the density functions ...