Maximum likelihood estimation for left-truncated log-logistic distributions with a given truncation point

TL;DR

This paper thoroughly analyzes the maximum likelihood estimation process for left-truncated log-logistic distributions with fixed truncation points, highlighting existence issues, solutions, and applications to real data.

Contribution

It introduces a criterion for the existence of MLE solutions, reduces the optimization to one dimension, and connects non-solvable cases to Pareto distributions, with practical simulation and testing methods.

Findings

MLE equations often lack solutions for small truncations

A criterion determines when a regular MLE solution exists

Pareto distribution is the limit for non-solvable cases

Abstract

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.