LogLogistic(a,b) - no Crystal Ball distribution
When Log(X) takes a Logistic distribution then X takes a LogLogistic distribution. Their parameters are related as follows:
EXP(Logistic(a,b)) = LogLogistic(1/b, EXP(a))
LogLogistic(a,1) is the standard LogLogistic distribution.
Uses
The LogLogistic distribution has the same relationship to the Logistic distribution that the Lognormal distribution has to the Normal distribution. If you feel that a variable is driven by some process that is the product of a number of variables, then a natural distribution to use is the Lognormal because of Central Limit Theorem. However, if one or two of these factors could be dominant, or correlated, so that the distribution is less spread than a lognormal, then the LogLogistic may be an appropriate distribution to try.
Comments
From the explanation of the Logistic distribution you can see that the limiting distribution of the geometric mean of the minimum and maximum samples from an exponential family distribution will take a LogLogistic distribution. There are relatively few applications of the LogLogistic distribution that directly use its underpinning mathematical model.
The parameter a defines the shape of the distribution, and the distribution's spread is proportional to b. Descriptions and applications of the LogLogistic model may be found in Bacon (1993), Diekmann (1992), Little, Adams, and Anderson (1994), Nandram (1989), and Singh, Lee, and George (1988).
The LogLogistic is also sometimes known as the Fisk distribution.
