Reduced bias estimation of the shape parameter of the log-logistic distribution

In the literature, the log-logistic distribution is commonly presented with two parameters: one that governs the shape of the model, and the other that governs its scale. However, to make this model more suitable for data analysis, an additional location parameter can be added, resulting in the three-parameter or shifted log-logistic model. In this paper, we introduce a new estimator for the shape parameter of a three-parameter log-logistic distribution that reduces bias. We also derive various properties of the proposed estimator. Additionally, a simulation study and an application example to a real data set are conducted to examine the efficiency for finite sample sizes. The theoretical and simulated results confirm that our proposed estimation method performs significantly better than other estimation methods found in the literature.