Percentile Double Ranked Set Sampling

In this paper, percentile double ranked set sampling (PDRSS) method is suggested for estimating the population mean. The PDRSS method is compared with the simple random sampling (SRS), ranked set sampling (RSS), median ranked set sampling (MRSS) and the extreme ranked set sampling (ERSS) methods. When the underlying distribution is symmetric, it turns out that PDRSS produce unbiased estimators of the population mean and it is more efficient than SRS, RSS, MRSS and ERSS based on the same sample size. For asymmetric distribution considered in this study, it is shown that PDRSS has a small bias and it is more efficient than RSS, MRSS and ERSS for most cases considered in this study.


INTRODUCTION
Ranked set sampling (RSS) is a cost effective sampling procedure when compared to the commonly used SRS in the situations where visual ranking of units can be done easily but the exact measurement of the units is difficult and expensive. McIntyre [4] proposed the sample mean based on RSS as an estimator of the population mean. He found that the estimator based on RSS is more efficient than SRS. Many modifications on RSS have been done since McIntyre [4] . Takahasi and Wakimoto [7] provided the necessary mathematical theory of RSS. Samawi and Muttlak [6] suggested RSS method for estimating the population ratio. Al-Saleh and Al-Kadiri [1] suggested double ranked set sampling method (DRSS) for estimating the population mean and they showed that the ranking at the second stage is easier than the ranking at the first stage. Muttlak [5] proposed percentile ranked set sampling (PRSS) for estimating the population mean. Jemain et al. [3] suggested multistage median ranked set sampling (MMRSS) for estimating the population median.

Ranked set sampling:
The RSS consist of selecting m random samples each of size m units from the population. The m units in each sample are ranked visually or by any inexpensive method with respect to the variable of interest. From the first set of m units, the smallest ranked unit is measured. From the second set of m units, the second smallest ranked unit is measured. The process is continued until from the mth set of m units the largest ranked unit is measured. Repeat the process n times if needed to obtain a set of size mn from initial m 2 n units.
Percentile ranked set sampling: In the percentile ranked set sampling (PRSS) procedure, select m random samples of size m units from the population and rank the units within each sample with respect to a variable of interest. If the sample size is even, select for measurement from the first m/2 samples the ( ( 1))th p m + smallest ranked unit and from the second m/2 samples the ( ( 1))th q m + smallest ranked unit, where 1 q = − p and 0 1 p ≤ ≤ . If the sample size is odd, select from the first (m-1)/2 samples the ( ( 1))th p m + smallest ranked unit and from the other (m-1)/2 samples the ( ( 1))th q m + smallest ranked unit and select from the remaining sample the median for that sample for actual measurement. The cycle may be repeated n times if needed to get mn units. These mn units are form the PRSS data.
Double ranked set sampling: The double ranked set sampling (DRSS) procedure can be described as follows: Identify m 3 units from the target population and divide these units randomly into m sets each of size m 2 . The procedure of ranked set sampling is applied on these sets to obtain m ranked set sampling each of size m, again apply the ranked set sampling procedure on the m ranked set sampling sets to obtain a DRSS of size m.
Percentile double ranked set sampling: To obtain a sample of size m based on PDRSS method, the following steps are required to be carried out: Step 1: Randomly select m 3 units from the target population and divide them into m samples each of size m 2 .
Step 2: Apply the RSS method on the m sets; this step yields m ranked set samples each of size m.
Step 3: Without doing any actual quantifications on the m sets obtained in Step 2, apply the PRSS method described above. Repeat the process n times to obtain a set of size mn from initial m 3 n units.
Note that if the sample size 3, m ≤ the percentile double ranked set sampling will be reduced to the usual ranked set sampling procedure. However, we will always take the nearest integer of the ( ( 1) with variance ( The estimator of the population mean for a RSS of size m (McIntyre 1952) is given by If m is odd, the PDRSSO estimator of the population mean is given by It is important to note that, using PDRSSE method when m is even, the measured units  If the distribution is asymmetric, the efficiencies are: For simulations, we assume that the cycle is repeated once. In Tables 1, 2 and 3, we summarized the results  for   7, 10, 11 m = respectively using RSS, MRSS, ERSS and PDRSS methods.
Based on Tables 1, 2 and 3, we can conclude the following:   and 11, the efficiency of PDRSS is 15.253 and 16.588 for estimating the mean of the standard normal distribution • For the normal and logistic distributions, PDRSS is more efficient than RSS and MRSS for all cases considered in this study. For the uniform distribution, PDRSS is more efficient than ERSS for most cases considered in this study • For asymmetric distributions considered in this study, PDRSS has a small bias. However, the MSE of PDRSS is less than the variance of SRS based on the same number of measured units.

CONCLUSIONS
In this paper, we suggested PDRSS method for estimating the population mean. Based on PDRSS, the findings of this paper may be summarized as follows: