Estimation of Location and Scale Parameters of Lognormal Distribution Using Median with Extreme Ranked Set Sampling

Ranked set sampling (RSS) is an effective method for data collection when direct measurements are difficult or costly, yet ranking through rough gauging is manageable. RSS assumes perfect ranking, an ideal often unattainable in practice, which can lead to reduced efficiencies. To address this issue, Muttlak (J. Appl. Stat. Sci., 6, 245–255, 1997) and Samawi et al. (Biom. J., 38, 577–586, 1996) introduced the Median Ranked Set Sampling (MRSS) and Extreme Ranked Set Sampling (ERSS) methods, respectively. Both provides unbiased estimators of the population mean and exhibit lower variance than traditional RSS and Simple Random Sampling (SRS), especially with symmetric distributions, although they may be biased in skewed distributions. Literature consistently favors RSS over SRS for parameter estimation in skewed distributions, despite RSS’s limitations, including the necessity of measuring each order statistic and the potential for ranking errors. To overcome these challenges, we propose a method that combines MRSS and ERSS, termed the median with extreme RSS (MERSS) method. MERSS with an odd sample size provides to facilitate easier identification of the median and extreme values and reduce ranking errors, thereby minimizing losses in relative precision. Our exploration of least squares estimation for the location and scale parameters of the lognormal distribution using MERSS reveals that the MERSS-based estimators, while not unbiased, significantly outperform SRS in terms of relative efficiency across all sample sizes. The method also perform better than RSS when estimating location parameters and quite close to RSS for estimating scale parameters. One real life example is demonstrated.