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

arXiv: https://arxiv.org/abs/2210.15155 · HTML: https://mlsanity.com/paper/2210.15155/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 27 Oct 2022.

## Built on by (most cited first, 1 of 1)

- [Bayesian Inference for Left-Truncated Log-Logistic Distributions for Time-to-event Data Analysis](https://mlsanity.com/paper/2506.17852/) (2025)

## Abstract (arXiv)

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 L1-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.