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arXiv · 2210.15155

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

Markus Kreer·Ayse Kizilersu·Jake Guscott·Lukas Christopher Schmitz·Anthony W. Thomas
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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.

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Markus Kreer, Ayse Kizilersu, Jake Guscott, Lukas Christopher Schmitz, Anthony W. Thomas. 2022-10-27. Maximum likelihood estimation for left-truncated log-logistic distributions with a given truncation point. https://arxiv.org/abs/2210.15155

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