Explicit expressions for the skewness and kurtosis are lengthy.
As tends to infinity the mean tends to , the variance and skewness tend to zero and the excess kurtosis tends to 6/5 (see also related distributions below).
It follows that the median is ,
the lower quartile is
and the upper quartile is .
The log-logistic distribution provides one parametric model for survival analysis. Unlike the more commonly used Weibull distribution, it can have a non-monotonichazard function: when the hazard function is unimodal (when ≤ 1, the hazard decreases monotonically). The fact that the cumulative distribution function can be written in closed form is particularly useful for analysis of survival data with censoring.
The log-logistic distribution can be used as the basis of an accelerated failure time model by allowing to differ between groups, or more generally by introducing covariates that affect but not by modelling as a linear function of the covariates.
The log-logistic distribution with shape parameter is the marginal distribution of the inter-times in a geometric-distributed counting process.
The log-logistic distribution has been used in hydrology for modelling stream flow rates and precipitation.
Extreme values like maximum one-day rainfall and river discharge per month or per year often follow a log-normal distribution. The log-normal distribution, however, needs a numeric approximation. As the log-logistic distribution, which can be solved analytically, is similar to the log-normal distribution, it can be used instead.
Finally, we may conclude that the Gini coefficient for the log-logistic distribution .
The log-logistic has been used as a model for the period of time beginning when some data leaves a software user application in a computer and the response is received by the same application after travelling through and being processed by other computers, applications, and network segments, most or all of them without hard real-time guarantees (for example, when an application is displaying data coming from a remote sensor connected to the Internet). It has been shown to be a more accurate probabilistic model for that than the log-normal distribution or others, as long as abrupt changes of regime in the sequences of those times are properly detected.
If X has a log-logistic distribution with scale parameter and shape parameter then Y = log(X) has a logistic distribution with location parameter and scale parameter
As the shape parameter of the log-logistic distribution increases, its shape increasingly resembles that of a (very narrow) logistic distribution. Informally:
The log-logistic distribution with shape parameter and scale parameter is the same as the generalized Pareto distribution with location parameter , shape parameter and scale parameter
The addition of another parameter (a shift parameter) formally results in a shifted log-logistic distribution, but this is usually considered in a different parameterization so that the distribution can be bounded above or bounded below.
Several different distributions are sometimes referred to as the generalized log-logistic distribution, as they contain the log-logistic as a special case. These include the Burr Type XII distribution (also known as the Singh–Maddala distribution) and the Dagum distribution, both of which include a second shape parameter. Both are in turn special cases of the even more general generalized beta distribution of the second kind. Another more straightforward generalization of the log-logistic is the shifted log-logistic distribution.
^ abEkawati, D.; Warsono; Kurniasari, D. (2014). "On the Moments, Cumulants, and Characteristic Function of the Log-Logistic Distribution". IPTEK, the Journal for Technology and Science. 25 (3): 78–82.