COMPUTATIONAL RESEARCH PROGRESS IN APPLIED SCIENCE & ENGINEERING (CRPASE) CRPASE: TRANSACTIONS OF APPLIED SCIENCES Journal homepage: http://www.crpase.com CRPASE: Transactions of Applied Sciences 7 (2) Article ID: 2350, 1–9, June 2021 ISSN 2423-4591 Research Article On Maximum Likelihood Estimators of the Parameters of Three-Parameters Weibull Distribution Using Different Ranked Set Sampling Schemes Essam Fawzy Aziz 1 , Mostafa Shaaban 2 1 Department of Statistics, Faculty of Commerce, Ain Shams University, Cairo, Egypt 2 The High Institute for Tourism, Hotels and Computer, El‑Seyouf, Alexandria, Egyp Keywords Abstract Simple Random Sampling, Ranked Set Sampling, Extreme Ranked Set Sampling, Double Ranked Set Sampling, Estimation Parameter. Al-Saleh and Al-Kadiri first proposed double rank set sampling (DRSS). It seems that this ranked set sampling (RSS) modification can reduce the loss of RSS efficiency caused by ranking errors, and it is more effective than RSS and simple random sampling (SRS) to estimate the population mean. The proposed likelihood function is used to estimate the parameters of the three-Parameters Weibull distribution. Based on double ranked set sampling, extreme ranked set sampling, ranked set sampling (RSS) and simple random sampling (SRS) designs, the maximum likelihood estimator (MLE) is compared with the corresponding likelihood estimator. A simulation was carried out and the absolute relative biases, mean square error (MSE) and relative efficiency of different schemes were compared. It is found that, MSEs based SRS data has the largest MSEs comparing to RSS and its modifications schemes. This study revealed that DRSS technique has the superior over the rest of other sampling schemes. In almost all cases, DRSS has the smallest MSEs and largest efficiencies. 1. Introduction The Weibull distribution is widely used in reliability and lifetime studies and proved an appropriate fit for most life data, except for data with non-monotonic empirical hazards. This type of data is often encountered in survival analysis, which makes it impossible for the Weibull model to analyze it. In many applications,  is assumed known (often  = 0), for which results [1] guarantee the existence of a unique maximum likelihood estimator , , and [2] the first of these is introduced tree parameter Weibull and concerned with asymptotic theory for maximum likelihood estimators. The cumulative distribution function (CDF) of the three-parameter Weibull distribution is given by (; , , ) = 1 −  −λ(x−α)  (1)  Corresponding Author: Mostafa Shaaban E-mail address: moushaaban@gmail.com Received: 2 June 2021; Revised: 13 March 2020; Accepted: 15 June 2020 https://doi.org/10.52547/crpase.7.2.2350 Please cite this article as: E. F. Aziz, M. Shaaban, On Maximum Likelihood Estimators of the Parameters of Three-Parameters Weibull Distribution Using Different Ranked Set Sampling Schemes, Computational Research Progress in Applied Science & Engineering, CRPASE: Transactions of Applied Sciences 7 (2021) 1–9, Article ID: 2350. where  > 0,  > 0 and  < . The parameters ,  and  are known as the scale, shape and location parameters, respectively. The corresponding probability density function (PDF) is (; , , ) = (x − α) −1  −λ(x−α)  (2) and the two parameters Weibull distribution is a special case of (2) when =1. 2. Some Ranked Set Sampling Techniques In this section, various sampling procedures for selection of units in the sample will be considered; brief descriptions of ranked set sampling (RSS), extreme ranked set sampling (ERSS) and double extreme ranked set sampling (DRSS) schemes will be introduced. McIntyre, 1952 [3] proposed Ranked Set Sampling (RSS) to improve the estimation of the population mean, and