Abstract—Stratified double extreme ranked set sampling (SDERSS) method is introduced and considered for estimating the population mean. The SDERSS is compared with the simple random sampling (SRS), stratified ranked set sampling (SRSS) and stratified simple set sampling (SSRS). It is shown that the SDERSS estimator is an unbiased of the population mean and more efficient than the estimators using SRS, SRSS and SSRS when the underlying distribution of the variable of interest is symmetric or asymmetric. Keywords—Double extreme ranked set sampling, Extreme ranked set sampling, Ranked set sampling, Stratified double extreme ranked set sampling. I. INTRODUCTION OONE denies the importance and the benefit of ranked set sampling method which was first proposed by McIntyre [11] for estimating the mean pasture and forage yields, and without proving he claimed that RSS X is an unbiased estimator for the population mean  , his method was developed and modified by many authors. Takahasi and Wakimoto [16] have established an accurate mathematical theory of ranked set sampling and they get the same results. Dell and Clutter [9] showed that the mean of the RSS is an unbiased estimator of the population mean, whatever or not there are errors in ranking. Samawi et al. [14] investigated the variety of extreme ranked set sample (ERSS) for estimating the population mean. Samawi [13] introduced the stratified ranked set sample method for estimating the population mean. Al-Saleh and Al-Kadiri [5] introduced double ranked set sampling for estimating the population mean. Samawi [15] suggested double extreme ranked set sample with application to regression estimator. Jemain et al. [10] suggested multistage extreme ranked set samples for estimating the population mean and they showed that the efficiency of the mean estimator using MERSS can be increased for specific value of the sample size n by increase the number of stages. For more about RSS see [4], [7], [12], [1] and [2], [3], and [8]. In this paper, new estimator using stratified double extreme ranked set sampling is suggested to estimate the population mean of symmetric and asymmetric distributions. The organization of this paper is as follows: In Section II, we present the stratified double extreme ranked set sampling. In Mahmoud. I. Syam is with Qatar University, Foundation Program, Mathematics Department, P.O.Box 2713, Doha, Qatar (Phone: 00974- 55468238; e-mail: M.syam@qu.edu.qa). K. Ibrahim, Jr., is with School of Mathematical Sciences, Universiti Kebangsaan Malaysia, 43600 UKM Bangi Selangor, Malaysia. A. I. Al-Omari is with Al al-Bayt University, Faculty of Science, Department of Mathematics P.O. Box 130095, Mafraq 25113, Jordan. Section III some notations and basic results are introduced. A simulation study is considered in Section IV. Finally, conclusions are introduced in Section V. II. STRATIFIED DOUBLE EXTREME RANKED SET SAMPLING In stratified sampling the population of N units is first divided into L subpopulations of L N N N , , , 2 1  units, respectively. These subpopulations are no overlapping and together they comprise the whole population, so that N N N N L      2 1 . The subpopulations are called strata. To obtain the full benefit from stratification, the values of the h N   1, 2, , h L   must be known. When the strata have been determined, a sample is drawn from each, the drawings being made in different strata. The sample sizes within the strata are denoted by 1 2 , , , L n n n  , respectively. The appropriate allocation of samples to different strata is very important in stratified sampling, in this article, the type of allocation method is proportional to stratum size. If a simple random sample is taken in each stratum, the whole procedure is described as stratified simple random sampling (SSRS). The stratified double extreme ranked set sampling (DERSS) is described as: Step 1. Identify 3 n elements from the target population and divide these elements randomly into 2 n sets each of size n elements. Step 2. For each set in step 2, if the sample size is even, select from the first 2 2 n sets the smallest ranked unit, and from the second 2 2 n sets the largest ranked unit. If the sample size is odd, select from the first 2 ) 1 (  n n sets choose the smallest ranked unit, and from the next n sets choose the median of each set, and from the other 2 ) 1 (  n n sets choose the largest ranked unit. This step yields n sets each of size n. Step 3. Apply the ERSS procedure again on the sets obtained from Step (2) to obtain a DERSS of size n. The cycle can be repeated m times if needed to get a sample of size nm units. Step 4. If the double extreme ranked set sample is used in each stratum, the whole procedure is described as stratified double extreme ranked set sampling (SDERSS). Estimating the Population Mean by Using Stratified Double Extreme Ranked Set Sample Mahmoud I. Syam, Kamarulzaman Ibrahim, Amer I. Al-Omari N World Academy of Science, Engineering and Technology International Journal of Mathematical and Computational Sciences Vol:9, No:4, 2015 228 International Scholarly and Scientific Research & Innovation 9(4) 2015 scholar.waset.org/1307-6892/10001069 International Science Index, Mathematical and Computational Sciences Vol:9, No:4, 2015 waset.org/Publication/10001069