  The maximum likelihood estimation of the left-truncated log-logistic
distribution with a given truncation point is analyzed in detail from both
mathematical and numerical perspectives. These maximum likelihood equations
often do not possess a solution, even for small truncations. A simple criterion
is provided for the existence of a regular maximum likelihood solution. In this
case a profile likelihood function can be constructed and the optimisation
problem is reduced to one dimension. When the maximum likelihood equations do
not admit a solution for certain data samples, it is shown that the Pareto
distribution is the $L^1$-limit of the degenerated left-truncated log-logistic
distribution. Using this mathematical information, a highly efficient Monte
Carlo simulation is performed to obtain critical values for some
goodness-of-fit tests. The confidence tables and an interpolation formula are
provided and several applications to real world data are presented.
