Investigating the Use of Stratified Percentile Ranked Set Sampling Method for Estimating the Population Mean

Stratified percentile ranked set sampling (SPRSS) method is suggested for estimating the population mean. The SPRSS is compared with the simple random sampling (SRS), stratified simple random sampling (SSRS) and stratified ranked set sampling (SRSS). It is shown that SPRSS estimator is an unbiased estimator of the population mean of symmetric distributions and is more efficient than its counterparts using SRS, SSRS and SRSS based on the same number of measured units.


Introduction
In last years, the ranked set sampling method, which was proposed by McIntyre (1952) to estimate mean pasture yields, was developed and modified by many authors to estimate the population parameters.Dell and Clutter (1972) showed that the mean of the RSS is an unbiased estimator of the population mean, whether or not there are errors in ranking.Al-Saleh and Al-Kadiri (2000) introduced double ranked set sampling for estimating the population mean.Al-Saleh and Al-Omari (2002) suggested multistage ranked set sampling that increase the efficiency of estimating the population mean for specific value of the sample size.Muttlak (2003b) suggested percentile ranked set sampling (PRSS) to estimate the population mean and showed that using PRSS procedure will reduce the errors in ranking comparing to RSS, since we only select and measure the pth or the qth percentile of the sample.Jemain and Al-Omari (2006) suggested double percentile ranked set sampling (DPRSS) for estimating the population mean and showed that the DPRSS mean is an unbiased estimator and more efficient than the SRS, RSS and PRSS if the underlying distribution is symmetric.Jemain and Al-Omari (2007) suggested multistage percentile ranked set sampling (MPRSS) to estimate the population mean, they showed that the efficiency of the mean estimator using MPRSS can be increased for specific value of the sample size by increasing the number of stages.For more details about RSS and its modifications see Al-Omari and Jaber (2008), Bouza (2002), Muttlak (2003a), Al-Nasser (2007) and Ohyama et al. (2008).
In this paper, stratified percentile ranked set sampling is suggested to estimate the population mean of symmetric and asymmetric distributions.This paper is organized as follows: In Section 2, some sampling methods are presented.Estimation of the population mean is given in Section 3. A simulation study is considered in Section 4. Finally, conclusions on the suggested estimator are given in Section 5

Sampling Methods
In stratified sampling method, the population of N units is divided into L non overlapping subpopulations each of N 1 , N 2 , ..., N L units, respectively, such that N 1 + N 2 + ... + N L = N .These subpopulations are called strata.For full benefit from stratification, the size of the hth subpopulation, denoted by N h for h = 1, 2, ..., L, must be known.Then the samples are drawn independently from each strata, producing samples sizes denoted by n 1 , n 2 , ..., n L , such that the total sample size is n = L P h=1 n h .If a simple random sample is taken from each stratum, the whole procedure is known as stratified simple random sampling (SSRS).
The ranked set sampling (RSS) is suggested by McIntyre (1952) can be conducted by selecting n random samples from the population of size n units each, and ranking each unit within each set with respect to the variable of interest.Then an actual measurement is taken of the unit with the smallest rank from the first sample.From the second sample an actual measurement is taken from the second smallest rank, and the procedure is continued until the unit with the largest rank is chosen for actual measurement from the nth sample.Thus we obtain a total of n measured units, one from each ordered sample of size n and this completed one cycle.The cycle may be repeated m times to obtain a sample of size nm units.
The percentile ranked set sampling (PRSS) procedure is proposed by Muttlak (2003b).The PRSS can be described as: select n random samples each of size n units from the population and rank each sample with respect to a variable of interest.If the sample size n is even, select for measurement from the first n/2 samples the p(n + 1)th smallest ranked unit and from the second n/2 samples the q(n + 1)th smallest ranked unit where 0 ≤ p ≤ 1 and p + q = 1.If the sample size n is odd, select for measurement from the first (n − 1)/2 samples the p(n + 1)th smallest ranked unit and from the last (n − 1)/2 samples the q(n + 1)th smallest ranked unit, and the median from the middle sample.The cycle can be repeated m times if needed to get a sample of size nm units.Note that we will always take the nearest integer of p(n + 1)th and q(n + 1)th.

Estimation of the population mean
The SRS estimator of the population mean, µ, based on a sample of size n is given by The estimator of the population mean for a RSS of size n is given by with variance where µ (i) is the mean of the ith order statistics, X (i) of a sample of size n.
In the case of stratified percentile ranked set sampling (SPRSS), when n h is even, the estimator of the population mean is defined as where W h = N h N , N h is the stratum size, N is the total population size and p + q = 1.
The variance of SPRSS1 is given by V ar V ar When n h is odd, the SPRSS estimator of the population mean is given by and the variance of SPRSS2 is ´+ V ar where µ h(p) and µ h(q) are the means of the order statistics which correspond to the pth and qth percentiles, respectively.Since the distribution is symmetric about µ, then µ h(p) + µ h(q) = 2µ .Therefore, we have ´+ where µ h(p) is the mean for the pth percentile in the first 2 ´samples in stratum h.µ h(q) is the mean for the qth percentile in the last ples in stratum h.µ h is the mean for the stratum h.Since the distribution is symmetric about µ, then we have µ Proof: If the sample size is even, the variance of The proof is the same for odd sample size.

Simulation study
In this section, a simulation study is designed for symmetric and asymmetric distributions with samples of sizes n = 7, 12, 14, 15, 18 to compare the SPRSS with the SRS, SSRS and SRSS methods.Without loss of generality, we assumed that the population is partitioned into two or three strata.Using 100000 replications, estimates of the means, variances and mean square errors are computed.The efficiency of SPRSS relative to SRS, SSRS and SRSS when the parent distribution is symmetric is given by ´, M = SRS, SSRS, SRSS, and when the distribution is asymmetric ´, where MSE is the mean square error.
In Table 1 to Table 6 we summarized the efficiency values of the estimators, while in Table 7 the bias values of the estimators for the mean of asymmetric distributions considered in this study are presented.