Efficiency comparison of maximum likelihood estimation in log–logistic distribution using median ranked set sampling

Abstract

This paper investigates maximum likelihood estimation (MLE) of the scale parameter, denoted as $\alpha$, and shape parameter, denoted as $\beta$, in the context of the log–logistic distribution, employing median ranked set sampling (MRSS). The study examines the scenarios where one of the parameters is known and cases where both parameters are unknown. The derived estimators based on MRSS are compared with conventional estimators in simple random sampling (SRS) and ranked set sampling (RSS), evaluating biases, mean squared errors, and relative efficiencies across various set and cycle sizes. Closed-form expressions of the Fisher information concerning the unknown parameters are obtained using the Mellin transform. A Monte Carlo simulation study is conducted using R software with 10,000 repetitions. Results indicate that when $\beta$ is known, the MLE of $\alpha$ based on MRSS demonstrates the highest efficiency, whereas when $\alpha$ is known, the MLE of $\beta$ based on RSS exhibits superior efficiency. In cases where both parameters are unknown, the MLEs of $\alpha$ and $\beta$ based on MRSS and RSS outperform those obtained through SRS.